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II  \v       I  \i;       IF  TIM-:  MOON; 


I'll  I-: I  I;    APPLICATION 

• 


-  i  \i 


IDO^ 

4IU6-1958 


INVESTIGATION 


rvv 

>N3\ 


•> 


CORRECTIONS  TO  IIANSEN'S  TABLES  OF  THE  MOON; 


TAltl.F.S   FOR  THK1R   A  1'1'MC  VI'ION. 


SIMON    NE\VCOM1?, 


IV  rt.  HAW. 


^ 

FORMING  PART  III  OF  PAPERS  PUBLISHED  BY  THE  COMMISSION  ON  THE  TRANSIT  OF  VENUS. 


WASHINGTON: 

GOVERNMENT     PRINTING     OPPICK. 
1876. 


TABLE    OF    CONTENTS. 


I'M.. 

|M>                      >   MOTE 5 

#  I.— INVESTIGATION  OF  ERRORS  OF  LONGITUDE. 

I    v.      I  •    .11 S 

V.uiulion g 

Mean  error  of  uliul.u  n^hl  ascension  al  different  limes  of  day q 

Value  of  solar  parallax  employed <> 

LiM  of  com-,  nous  1,1  ••  Argument  Fondamcnl.il"  . 10 

General  ideas  which  foim  the  basis  of  this  invcMigation 11 

Differential  coefficients 12 

Mean  apparent  error  of  1 1  arum's  Table*  in  right  ascension 12 

Su.Mcn  apparent  alteration  in  mean  motion  of  the  moon 13 

<'..ru-.  n. in*  |.>r  limb  and  observatory  to  tender  observation*  siiirtlr  comparable 13 

Mean  outstanding  tabular  error  of  the  moon  in  longitude  .  . 13 

'rctkms  of  short  period  actually  applied 14 

K<juaiion  connecting  errors  of  moon's  tabular  right  ascension  with  emu*  <>t  lunar  element*.  .......  id 

-  of  errors  of  moon's  i  <>u<-.  i«-.l  n^hl  jurcnviin  given  liy  observations  al  Greenwich  and  Washington.  ...  17 

s  -tnal  equation*  d>r  ilnrrmining  M,  4,  and  t  by  least  squares :•, 

V .tlucs  of  outstanding  errors  of  lunar  elements  for  t-.K  M  \  t ." 20 

Apparent  periodic  character  of  the  corrections  to  the  ecccnlm  ii\  .md  perigee 20 

Formula!  for  the  new  inequality  of  longitude 24 

Discussion  of  Greenwich  observations  of  the  moon  from  1847  to  iSjg ...  94 

Sums  of  residual  errors 26 

••ctions  to  pcccnlriciiv.  longitude  of  perigee,  and  mojn'»  longitude 39 

g  a.— INVESTIGATION  OF  POLAR  DISTAM  I    AM)  LATITUDE. 

Corrections  to  declination  depending  on  errors  of  longitude 32 

Constant  corrections  to  reduce  declinations  to  same  fundamental  standard 32 

Sums  of  errors  of  moon's  corrected  declination,  giren  by  observations  at  Greenwich  and  Washington 34 

Correction  to  inclination  of  orbit  and  longitude  of  node 36 

£  3.— AUXILIARY    TABLES   FOR    FACILITATING    THK  COMPUTATION   OF    THE  CORRECTIONS  To 

IIANSEN'S  "TAHI.KS  DK  LA  LI'NE". 

Summary  of  corrections  to  mean  and  true  longitude  of  the  moon  from  Hansen's  Tables 37 

K\|>lanalion  of  tables  for  applying  these  corrections 37 

Example  of  the  use  of  the  tables 40 

•  lions  to  the  Ephemcris  derived  fiom  llansen's  Tables  of  the  Moon,  for  Greenwich  mean  noon  of  each  day, 

from  1874.  September  I,  to  1875.  January  31 41 

Tables  I.  II.  Ill,  the  arguments  g,  I*.  A.  B,  u 4j 

Tables  IV,  V,  VI,  secular  and  empirical  terms 46 

Table  VII.  terms  of  mean  longitude 47 

Talilc  VIII.  u-rmsof  true  longitude 4$ 

Tables  IX.  X,  factors  for  reduction  to  longitude  in  orbit ;  and  for  correction  of  latitude  and  reduction  to  ecliptic 

longitude Jo 

Table  XI.  factors  for  convening  small  changes  of  longitude  and  latitude  into  changes  of  right  ascension  and  decli. 

nation «i 


INTRODUCTORY    NOTE. 


When  (lie  problem  <»!' utilizing  tlie  observations  of  occultatiuni*  nt  tin-  sevenil  Transit 
of  Venn*  stations,  so  as  to  determine  the  longitudes  of  those  stations  with  all  attainable 
aeeur.iey,  was  presented  tu  tin:  Commission  on  tin-  Transit  of  Venus,  it  was  found  neces- 
sary t«>  make  i  cairl'ul  .l.-t.-i  ininatiun  ol  tlic  errors  of  tlie  lunar  e|iliemeris  l>efon-  an 
i-ntin-lv  satisfactory  solution  <»(  tin-  |>roli|<-m  nml.l  In-  att«-in|iti-i|.  Tlie  Si-en-tary  of  tin- 
( '..niinissioii  was  therefor*-  eharijed  with  this  work,  most  of  the  eom|iiitations  on  whirli 
have  l»een  made  under  his  ilin-elion  l>y  Mr.  1>.  I*.  To«hl,  c-ompulcr  for  tin;  Commisrtion. 

WASIIINOTOM,  May  25,  1876. 


CORRECTIONS  TO   BE  APPLIED  TO  HANSEN'S  TABLES 

OF  THE   MOON. 


INVI--I  H.  \  I  I.  -\   OF  ERRORS  OF  I.ONC.ITUDK. 

One  of  the  must  important  operations  in  connection  with  the  nhserval inns  nt  I  lie 
Iraii-il  nl  Venus  is  (In-  accurate  determination  of  the  longitude!  of  the  stations.  Many 
nt'  these  st;iliniis  arc  so  fur  reim.w-d  from  telegraphic  communication  that  tin-  longitude! 
mn-t  .l.-|i.-n.l  mainly  on  tin-  moon.  Determinations  nl  longitude  from  moon  culmina- 
tiuns  arc  lound  l»y  ev|»erience  to  lie  subject  In  constant  errors  which  it  is  difficult  l«» 
ilctcriniiic  and  allow  for.  It  was  therefore  a  part  nl'  the  policy  of  the  American  < 'oni- 
ini-sion  to  ilepenil  on  occnltatinns  rnllicr  than  ii|»ni  UMNIII  culminations  for  the  determi- 
nation nt  lnnt;itiiiles.  The  reason  for  this  course  is,  that  I  lie  disappearance  of  a  >tar 
liehiml  the  limli  of  the  moon  is  a  sudden  phenomenon,  the  time  ol'  which  can  always  In- 
fixed within  a  fraction  of  a  second.  If  the  cphemeris  of  the  moon  and  star  were  cor- 
rect, and  the  disk  of  the  former  a  |>erfect  circle,  the  longitude  could  lie  determined 
from  the  occultation  with  the  same  decree  of  accuracy  that  the  phenomenon  could  he 
oliM-rved.  The  question  arises,  how  far  these  sources  of  error  can  be  dimininhed.  The 
inequalities  of,  I  IIP  lunar  surface  form  a  source  of  error  which  it  is  im|mssilile  to  avoid, 
luit  \\hich  is  comparatively  innocuous  when  many  observations  are  made,  since  lin- 
en-ore will  lie  purely  accidental,  and  will  then-fore  lie  eliminated  from  the  mean  nl  a 
great  number  of  observations. 

The  position  of  (he  star  can  be  determined  by  meridian  observations  with  almost 
any  required  degree  of  accuracy.  We  have.  then,  only  to  see  how  far  the  ern>rs  of  the 
lunar  ephemeris  can  be  diminished:  and  to  reduce  these  errors  to  a  minimum  is  the 
object  ol  the  present  paper. 

Han-en's  tables  are  taken  for  this  pur|M>se.  because  there  is  reason  to  believe  that 
the  perturbations  on  which  these  tables  are  founded  are,  in  the  main,  extremely 
accurate;  more  accurate  and  complete,  in  fact,  than  any  others  which  have  been 
tabulated.  Still,  before  they  can  be  used  lor  the  purpose  in  question,  a  number  o! 
important  corrections  are  required,  which  we  may  divide  into  twoclas-e-. — corrections 
to  the  theory,  and  to  the  element*. 

It  is  well  known  that  llansen  incn-a.-ed  all  I  lie  perturbations  of  his  table.-  by  the 
r.in-tant  liictor  O.OOOI 544.  on  account  of  a  siip|Ni>ed  \\ant  of  coincidence  bet u ceil  the 


8 

Center  of  figure  and  the  center  of  gravity  of  the  moon.  I  have  shown  that  Hanscn 
fails  to  sustain  this  position,  and  that  there  is  no  good  reason  to  suppose  that  the  moon 
differs  from  any  other  of  the  heavenly  bodies  in  this  respect.*  Our  first  course  would 
therefore  be  to  diminish  all  of  Hanson's  inequalities  by  this  factor,  were  it  not  that  there 
are  reasons  why  each  of  the  two  greatest  perturbations  of  the  moon's  motion, — the  eve'c- 
tionand  the  variation, — should  be  found  larger  from  observation  than  he  found  them  from 
theory. 

Evection. — The  evection  has  the  eccentricity  as  a  factor;  the  value  of  the  other 
factor  being  nearly  0.4.  If,  then,  the  adopted  eccentricity  of  the  moon  be  erroneous, 
the  computed  evection  will  be  erroneous  by  four-tenths  the  amount  of  the  error.  Now, 
by  reference  to  Hanson's  "Darlegung  dcr  iheoretischcn  Berechnuttg  der  in  den  Mondtu- 
feln  angnrandten  Storungcn'^  (page  173),  it  will  be  seen  that  the  eccentricity  adopted 
throughout  in  the  computation  of  the  perturbations  of  the  moon  is  less  by  0.0000073 
than  the  value  lie  finally  found  from  observation,  and  adopted  in  the  tables.  Had  he 
used  the  latter  value,  the  theoretical  evection  would  have  been  greater  by  the  fraction 

— -^  =0.000133.  The  factor  actually  used  being  0.00015 44,  ^'e  evection,  thus  in- 
•°549O°" 

creased,  is  too  large  by  only  0.00002 1   of  its  entire  amount,  or  o".O9.     Consequently 
the  tabular  coefficient  of  evection  should  be  diminished  by  this  amount.     Precisely  the 
same  result  follows,  if  we  adopt  Hausen's  view  of  a  separation  of  the  centers  of  figure 
and  gravity  of  the  moon;  and  Ilansen  himself  is  led  to  it  on  page  175  of  the  work  cited, 
only  instead  of  o".09,  he  says,  "kein  voiles  Zehntheil  einer  Secunde.'1 

Variation. — That  the  coefficient  of  variation  resulting  from  meridian  observations 
will  be  greater  than  the  actual  coefficient  may  be  anticipated  from  (he  following  con- 
siderations. The  inequality  in  question  attains  its  maxima  and  minima  in  the  moon's 
octants.  In  the  first  octant,  we  have  a  maximum.  The  elongation  of  the  moon  from 
the  sun  is  then  about  3h;  and  the  observed  position  of  the  moon  is  mainly  dependent  on 
observations  of  the  first  limb  made  in  the  day  time,  when  the  apparent  semi-diameter  of 
the  moon  will  be  diminished  by  the  brilliancy  of  the  surrounding  sky.  No  account  of 
this  diminution  of  the  apparent  semi-diameter  being  taken  in  the  reductions,  the  semi- 
diameter  actually  applied  is  too  large,  and  the  observed  right  ascension  of  the  moon  is 
also  too  large. 

When  the  moon  reaches  the  third  octant,  the  value  of  the  variation  attains  its  min- 
imum. The  moon  then  transits  at  9*,  and  the  meridian  observation  is  made  on  the  first 
limb,  while  the  apparent  semi-diameter  is  increased  by  the  irradiation  consequent  upon 
the  contrast  between  the  moon  and  the  sky.  The  result  will  be  that  the  observed  right 
ascension  will  be  too  small. 

The  same  causes  will  make  the  observed  right  ascension  too  great  in  the  fifth 
octant,  and  too  small  in  the  seventh.  These  positive  and  negative  errors  of  observed 
right  ascension  correspond  to  the  times  of  maximum  and  minimum  effects  of  variation 
in  increasing  the  longitude  of  the  moon.  Therefore,  the  observed  variation  will  appa- 

•  Proceedings  of  the  American  Association  for  tlio  Advancement  of  Science,  1868.— Silliinan'a  American  Journal 
of  Science,  November,  1868. 

t  Abliaiidlmigen  dor  mrttli«.-ni»tinch-pliyni«clion  Cliume  der  Koniglich  SUohumchen  Genollschaft  der  \Vissenncuaften 
Hand  vi. 


9 

rently  l>c  larger    than    tlir  iictual  vaiiai \\liali-\rr    llii>    may  I"-.      This  seems  a  much 

more  natural  ami  |iriilial>l>-  eaii-e  lor  I  In-  apparent  excess  of  tin-  oliserved  over  the  theoreti- 
cal pertuiliation>  llian  that  assigned  liy  llanscii.  Hansen's  factor  onl\  increases  tin-  coelli- 
cient  in  question  liy  o  ..;;:  Imt  il  •.••••iii-  prolialde  tliat  tin- variation  derived  I'ruiii  oliser- 
\ation-  alone  \\onlil  In-  Ml  laii/i-r  than  ll.m-i  n'-  increased  \ariation.  In  tact,  in  1X67,  1 
toiniil.  li\-  comparin::  tin-  error-,  ol"  tin-  lunar  ephemeri>  when  the  moon  culminated  ut 
dilli-rent  tinif>  nl'  tin-  ilay,  lliat  tin-  ell'eet  of  tlic  yreater  irradiation  at  iii^lil  \\a>  \n\ 
stioiiijly  marked,  huriiiir  tin-  limr  JGUt  1862-65  the  mean  errors  ot  the  tallies  ill 
rijjlil  a«ct  IIMOII  at  ilillt-rfiit  tunes  ol'  day  u.-n-  a>  follows:* 

•. 

Itflort-  Min^rt  —  O.  154 

AHt-r  liri^'lit  tlayliirlit  in  tin- cvfiiinij.  .    — 0.093 

I'.,  ion-  liriylit  da\liirlit  in  tin-  morning.  .  .  .    -{-0.09! 
AlW  snnrisf  . ...  .    4-0.  15.; 

In  tin-  tlilli-rrncf  IM  tu. m  tin-  results  liir  cai-li  limli,  (In-  fllt-ct  ol'  incri-ascil  irratlia- 
titin  M'fins  to  IP.-  .  .'.06. 

'I'lir  only  i. -111.1111111^  I. -mi  \\  liich  i-  laiu1--  i-noiiL'li  Iti  In-  materially  aiTccli-d  li\  tlic 
UK T.MX.-  in  (|iu  xii.Hi  is  tlic  annual  ci|iiatioii,  ot  \\liicli  tin-  incrca>r  is  D".IO. 

.inrc  at  tlic  errors  of  I  lan>i-n'>  (allies.  yiM-ii  li\  mcritliaii  oli>ei  valions,  will  ti|i«>w 

that  th.-i  ii.nv   alitmt  the    til I    lirsl  quarter,  ami,  indeed,  dnrin<r  tlie  first  lialf  nf  the 

lunation,  an-  in  the  mean  less  \>\  l>et\\een  ;  and  4"  than  during  tlie  sc<-ontl  half. 
Hence,  either  the  >emi -diameter,  or  (lie  parallactic  ei|iiation,  or  liuth,  arc  ton  lari;e.  The 
parallaclic  ci|iialion  used  liy  Haiisen  corrcs|M)iuls  In  a  value  S". 916  for  the  solar  paral- 
lax, which  \alnc  is  too  lame  liy  (irolialdy  nut  much  less  than  o".lO.  The  result 
which  I  dedneed  in  iS(>7  from  all  the  really  \alualde  data  extant  was  S  .SjS;  ami  the 
ilelerminations  which  have  since  lieen  made,  u  hen  revised  witli  the  lies!  tlata,  seem  tu 
iiitlicate  a  diminution  of  this  \alne  rather  than  an  increase.  These  indication-,  are,  hnw- 
a  little  loo  intlelinite  to  predicate  aiiylliiiiir  II|NHI.  I  shall  then-tore  con- 
tinue to  W  (S,  which  will  diminish  llansen's  value  l.\  o".o68.  The  corrcs|ximliiiij 
diminution  in  the  princi|ud  parallaclic  term  will  lie  <>".(>f>,  \\  liile  there  will  he  two  other 
term-  to  reeeive  a  smaller  diminution. 

• 

This  correclion  will   slill  leave   a   ilillerenee  ol'  admit    2"    lie! \\een  the  n>ull>  from 

the  fust  and  set I  limli-,  w  liicli  will  lie  accounted  for  liy  an  error  of  i"  in  the  adopted 

semi-diameter.      This  correction  tu    the  semi-diameter  is  //  ]>n<ni  quite  prolialde.  as  the 
improved  meridian  instruments  of  the  present  time   i:ive  a  semi-diameter  of  the  sun   i' 
leas  tlian    tlie   ulder   ones    from  which    the   diameters  adopted  in  our  ephemerides  were 
derived.      It    is  to    lie   expected    ( hat  I  he  semi-diameter  of  t  lie  moon  will  exhiliit  a  sim- 
ilar apparent   diminution. 

I'icima  note  in  llans.-n'-  /'.// 1<  -///'^  (page  439),  il  will  lie  seen  that  tine  of  I  lie  term> 
in  the  true  lon-itude  has  crept  into  the  (aides  with  a  wrunir  siyn.  Asemploved  in  the  talde-. 
and  «i\en  mi  pa^'e  i^ufthe  in!  rutliict  ion.  it  is,  +o"-335  Kin  (  2«  —  4  gf  -|-  20)—  4&''). 

As   revised   ill   the  l>,lll,  L'HHZ.  it   is  .  —  O".28S8ill 

Therefore  the  taldes  neet)  the  correction       ...    —  o".62    sin 

•   l,n.-t, _;.,!,..„   ..I    I!,.    I  leaner  or   II..    >.»..!• 

IB 


10 

The  following  is  a  list  of  the  corrections  \vc  have  so  far  deduced  to  Hansen's  (allies. 
They  should  in  strictness  be  applied  to  the  mean  longitude,  or  "Argument  fwtliamfntai", 
but  they  may  without  serious  error  be  applied  to  the  true  longitude. 

Put 

D,  the  argument  of  parallactic  inequality,  or  mean  elongation  of  the  moon  from 

the  sun  ; 

£,    the  moon's  mean  anomaly ; 

g',  the  sun's  mean  anomaly  ; 

co,   the  distance  of  the  moon's  perigee  from  the  ascending  node; 

co',  the  distance  of  the  sun's  perigee  from  the  same  node. 

We  then  have 

D  =  g  —  g'  +  &  —  oaf, 

and  the  corrections  in  question  are, 

// 
+  0.96  sin   D  ^ 

-f-  O.O7  Sin   (/? g    )      >  rarallaetic  term* 

-  0.13  sin 

-4-  O.OQ  8111  £,' 

^—0.33  sill   2   D  Variation, 

—  O.  I  O  si  II   (  2  D g)        Ktntiim. 

—  O.62  sill  (  g  2  —  4  s'  -\-  2  Csl  —  4  ft)')     Accidental  eirnr. 

The  fourth  and  fifth  terms  of  this  expression  have  the  effect  to  remove  the  increase 
which  Hanseii  applied  to  his  inequalities  on  account  of  the  position  of  the  center  of 
gravity  of  the  moon,  while  the  sixth  is  the  result  of  the  slight  error  of  the  eccentricity 
which  he  employed  in  computing  the  coefficient  of  evection. 

In  comparing  with  meridian  observations  which  have  been  reduced  without  any 
correction  to  the  apparent,  semi-diameter  depending  on  the  time  of  day,  the  correction 
of  variation  may  also  be  omitted,  since  a  yet  larger  apparent  correction,  having  the  oppo' 
site  algebraic  sign,  will  result  from  the  apparent  variations  of  that  semi-dinmeler,  as 
already  explained. 

As  regards  the  possible  corrections  to  the  elements  of  Hansen's  tables,  it,  is  to  be 
remarked  that  thai  investigator  did  not  avail  himself  of  the  elements  of  the  lunar  orbit 
deduced  by  Airy  from  the  (Greenwich  observations  between  1750  and  1830,  but  obtained 
his  final  values  of  the  elements  by  a  comparison  of  his  own.  Of  the  nature  and  extent 
of  the  observations  tlius  employed,  we  have  no  details;  but  it  is  not  likely  that  more 
than  a  very  small  fraction  of  the  entire  mass  of  observations  was  used,  and  it  can  there- 
fore hardly  be  expected  that  the  elements  were  determined  with  (lie  hist  degree  of 
accuracy.  Any  error  in  the  motion  of  the  perigee  or  node  will  constantly  increase  with 
t lie  time.  If,  in  addition  to  this,  we  relied,  that  the  meridian  observations  of  the  last 
twenty  \  ears  are  tin-  more  accurate  than  (hose  Hanseii  had  at  his  disposal,  it,  will  not 
seem  at  all  surprisinj,'  to  find  quite  sensible  errors  in  the  present  longitudes  of  the  lunar 
periirer  and  node  as  derived  by  Hanseii.  Our  ne\<  step  will  therefore  be  to  determine 


II 

\v  li  lion-    t.i   ||.uiNen'>    element*    art-  indicated    liy  tin-  recent  olisei  \.i!  HUI*  of  the 

union    111. nli-    ;it  (  i  reen\\  irli    ami   Washington    since     1 863,    .1    peiind    dmini:    which    Imth 
.,|    oli-ervatioiis  an-  i  aii-fully  compared  \\itli    Han-en'-  laldes. 

Tin-  irem-ral    ideas  mi  which  tin-  pies. -lit  inxexii-aii I'  these  corrections  i*   liMed 

lli.'  error*  of  the  's  laliular  longitude  an-  of  l\\u  class.-...      It  progressive 

correction,  \\liicli  apparently  increases  uniformly  w ith  the  time;  and  emus  of  whorl 
period,  tin-  principal  ones  nl'  which  -jo  llinniu'li  their  period  durini:  "lie  n-vnlnlion  nl'  tin- 
in. i. >n  in-  lt-».  In  ilcliTiniiiinx'  III'1  'Tfiii-N  nl  tin-  lir.-l  fliws  frnni  oliMTvalinii.  llinsc  nl' 
the  MTiuiil  rla--  ma\  !•••  i.-^anl.'.l  a>  acciilciilal  i-rrnr>,  llif  ••Hi-cl  nl  which  \\ill  \te  »'lim- 
in. iic  . I  in. 111  (lie  mi-ail  u|  .1  laii.'<-  iiiiinlirr  nl'  uli-rn al loiis.  Since,  in  a  series  nl°  nUserva- 
tinti>  i-\iriiiliiiir  tlimnuli  a  iniiiilier  nl'  \eais.  (lie  maxima  ami  minima  nl  each  term  of 
»imrt  |terin<l  \\ill  fall  iinliscriminateU  mtn  all  |iarts  nf  all  the  other  |»erinils,  each  periodic 
eiirreetiiin  may  lie  uetermined  as  if  the  ell'ecls  ofllir  «itliei>  \\en-  jiuiely  acciilental 
•  MI.''-  At  (he  Mime  time,  as  the  elimination  of  each  periodic  error  Irom  the  maxima 
.iiul  minima  nl'  all  the  nlheo  cannot  Ke  complete  in  any  finite  time,  it  is  desiralile  that 
each  |H-rioilic  cnrrectinn  nl  sensilile  maumhnle  «  hicli  v\  e  can  determine  lielorehand  shall 
lie  a|i|died  to  the  residuals  liefore  the  latlei  are  Used  to  determine  the  cnrrectinlls  to  the 
element-. 

The  corrections  of  the  elements  of  longitude  have  lieen  made  to  de|iend  |n  incipall y 
UJMIII  tin-  oli»--r\ed  riuht  a.-censinns,  instead  of  reducing  ihe  oliser\ed  errors  of  i i^'hl 
. •nsimi  and  polar  distance  to  errors  of  longitude  and  latitude.  The  lea-mi  lor  this 
coiir-i  IN.  iiul  the  a|iparent  emus  of  polar  distance,  after  correcliiiu'  them  approximately 
fnr  errors  of  the  elements  easily  determined,  u  ill  aiise  principally  from  error*  of  ..!•• 
\alinn,  and  not  In  MM  ermrs  nf  the  tallies.  In  fad.  the  oliset  \ations  of  the  IIIIHIII'S  declin.i 
lion  an-  sometimes  allected  with  accidental  errors  of  a  magnitude  \\hieh  it  is  ditlicult  to 
account  for,  especially  in  the  case  of  Wa>hini;lon.  (iraiitiny  that  Ihe  moon  move*  in  a 
plain-  the  |M>sition  of  which  can  lie  very  accurately  determined,  we  liave  aOerward  only 
to  determine  the  moon's  |M>sitioii  in  that  plane,  and  this  can  lie  done  from  an  olisened 
right  aM-ensioii  almost  as  well  ax  if  we  had  a  direrlly  olisened  loogittlda  The  lonyi- 
tude  thus  determined  will  In:  less  likely  to  lie  allected  \\ilK  systematic  errors  thai)  if  we 
siip|Hise  the  position  entirely  unknown,  and  change  (hi-  errors  of  rinht  ascensi(»n  and 
declination  In  errors  of  loimitude  and  latitude,  without  regard  to  the  |MissiMe  constant 
errors  nf  the  measured  ilerliiiations. 

Formula-  for  expres>int:  (he  longitude  and  latitude  nf  the  moon  in  terms  of  the 
lunar  elements  are  liiven  liy  llansen  in  a  piistlinnioiis  memoir.*  The  following;  ti-rm- 
are  siitlicient  for  our  present  purpi- 

I'ut 

/,         tin-  mnmi's  Imiiritiide  ill  mliit  ; 
6,       the  lnni;itnile  of  the  ascendini;  node; 
*,        the  inclination  of  the  nrliit  to  the  ecliptic: 
\  the  moon's  ri^'lil  ascension  and  declination: 
tin-  oliliiplity  ot'   the  ecliptic. 

•  I  ..l«-i  .1..-  HarmU-lluiii;  .1.  r  _-t  ..|.  :,    \  ,.I>II-IKIIIIK  nn.l  AbwrirtiunK  do  M  :  UMtiM  ttt  lAff  !•  4cT B«bo 

and  .1,  r  KnoUnUoRr.     Abli»».llui^"ii  drr  Kopiglinh  BlihriMhiB  O«»IUeb»ft  dcr  WiweuKlicArD,  lid.  i,  Vo,  Till. 


12 


We  then  have,  approximately, 

a  —  I—  2°.5sin  2/—  i°.i  sin(2/  —  (9) -f  i°.i  sin  0 
sin  S  —  sin  oo  sin  I  +  cos  ca  sin  i  sin  (/  —  0) 

—  0.40  sin  /  -f  0.08  sin  (7—  <9) 
The  differential  co-efficients  derived  from  these  expressions  are, 

—  i  —  0.037  cos  (2  /  —  0)  —  0.087  cos  2  / 
dl 

''"'  —  o.oi8cos0+ooiScos(2/  —  9) 
°^-  —  0.2 1  sin  9  —  0.2 1  sin  (2  /  —  9) 


cos  S 


cos 


—  0.40  cos  /  -f-  0.08  cos  (/  —  0) 

=  (0.40  -f  0.08  cos  0)  cos  I  +  0.08  sin  9  sin , 

=  —  0.08 1  cos  (I  —  9} 


dS 


cos  S  -      —  0.92  sin  (/  —  <9) 
r/t 

From  the  first  three  ibrmula1,  it  will  be  seen,  that  the  mean  error  in  right  ascension 
is  very  nearly  the  same  as  the  mean  error  in  longitude ;  (lie  periodic  corrections  being 
supposed  to  lie  eliminated  from  this  mean. 

The  investigation  of  the  corrections  from  observations  is  now  made  as  follows: 
All  the  apparent  errors  of  the  tables  derived  from  the  meridian  observations  at  Green- 
wich and  Washington  since  I  S62  have  been  collected,  arranged  in  the  order  of  dates 
and  the  mean  taken  for  each  year;  observations  of  the  separate  limbs  being  kept  sepa- 
rate. The  mean  error  in  right  ascension  for  each  year  is  as  follows: 
Apparent  errors  of  Hanson's  tables  in  ft.  A. 


•    Greenwich. 

• 

Washington. 

Mean. 

"Year. 

L 

II. 

Diff. 

I. 

II. 

Dill. 

I. 

II. 

Mean. 

1862 

'• 

II 

» 

•• 

» 

" 

-  3-6 

—  0.6 

—    2.1 

1863 

—  2.3 

+  0.5 

--  o.o 

1864 

—    I.O 

+  1.3 

+  0.4 

1865 

—    0.2 

+  3-o 

3-2 

+  0.3 

+  3-9 

3-6 

o.o 

+  3-4 

+  1.7 

1866 

+    1.2 

+  3-6 

2.4 

+  0.9 

+  4-5 

3.6 

+    I.O 

+  4-0 

+  2.5 

1867 

+    2.4 

+  5-7 

3-3 

+  2.4 

+  5.8 

;-4 

+  2.4 

+  5-8 

+  4.1 

1868 

+    2.6 

+  6.0 

3-4 

+  2.4 

+  6.6 

4.2 

+  2.5 

+  6.3 

+  4-4 

1869 

+    3-3 

+  5-6 

2.3 

+   ? 

+  7-4 

4.0 

+  3.4 

+  6.5 

+  4-9 

1870 

+  3-4 

+  6.6 

3-2 

+  4-6 

+  72 

2.6 

+  4-0 

+  6.9 

+  5-4 

1871 

+  5-4 

+    8.2 

2.8 

+  5-1 

+   7-8 

2-7 

+  5.2 

+  8.0 

+  6.6 

1872 

+  6.0 

+  8.7 

2.7 

+   6.2 

1      '!.<> 

3-4 

+  6.1 

+  9.2 

+   7-6 

1873 

+  6.9 

+  9-4 

2-5 

+  6.9 

+  10.2 

3-3 

+  6.9 

+  1O.  2 

+   8.6 

1874 

+  8.1 

+  11.  4 

3-3 

+  7.1 

+  10.8 

3-7 

+  7-6 

+  11.  1 

+  9-4 

The  last  column   exhibits  the  apparent  tabular  errors  in  mean  right  ascension,  and 


L8 


the  if loi  f  in  ii  it -.in  longitude.  ;i>  .If  IIM  d  f.u-h  year  I  mm  .ill  tl Itserxatmns.      The  siulilfii 

.I|>|MI.  nl  .ill.  i.iin. M  nl  in  .iily  inn-  M-fiiinl  per  .ininiiii  in  the  mean  motion  oi  the  moon, 
exhiliiled  in  (Ills  Ciilliinii,  seems  tu  nif  mif  of  tin-  must  f\l  r.ionliii.ii  y  of  astronomical 

plici if n.i  :    luit,  ;IN  |   II.IM-    diM-iis*.,-,!    il  in    M-MI.I!    p.i|>fi-    .luring  the  l;i>t   I'm-  yearn,  I 

shall  <ln  mi  iiinif  here  than  rail  attfiitimi  to   its  continuance,  .in. I  to  the  ini|Hissiliility  of 

lepr.-M-ntiii;,'  ii  liy  any  MIL ill  number  o(  )••  liuilu-  (elms  w itliont  introducing  diaoonkuM 

int. i  tin-  lonu'ilude  during  prc\  ion- 
It   will  In-  seen  that  then-  an-  iliscordaiief.N  lietweeii  tin-  re>nlts  of  tin-  t\\o  uli>fr\a- 

loin  >,  > •liiiif-  aiiiiinniiii^  tu  IIHIIV  than  a  >i-rnii,l.      In  ilfifiiiiiiiin^  tin-  roncftiuns  <if 

short  |ifiii>il.  it  i>  ilf-iralili:  tu  rciluri-  tin-  s\  >tnn.ilif  fimrs  c  \lf  mlinir  tlmillgli  rach 
M-at  to  a  minimum;  lln  i|iif-lion  whctlifi  such  rrmrs  arc*  in  the  theory  or  the  ulisfrva- 
tiuiis  linn;,'  imlillf  rent.  It  i>  also  ilcsiralilf  that  in  taking  the  mean  of  the  results  of  the 
two  oliM-i  vatorifs,  they  .-liouM  lie  made  eomiKiralili1  with  each  other  liy  correct  iny  either 
uf  them  I'ur  the  s\ste in;ni<-  ilillfrencf.  These  currrctions,  of  eom>f,  only  adinil  ul 
a|'|iro\imate  ih- termination,  .mil  they  II.IM-  In-en  applied  eat  h  year  tu  that  olisi-rsatur)'  or 
that  limli  ot'the  moon  in  whtcb, judging  lioni  the  di-vuitions  Iruin  iinitiirin  |iru^ressiun,  il 
\\as  jiid^'fd  most  likely  that  the  disrurdaiiee  existed.  The  foOowing  are  the  eorreetions 
a«-tuall\  ;i|»|ilied  to  the  >••  \.-r.il  ela.-si-s  ul'  l.il.nl.ir  f 


(•II  I   Mil  U   II. 

U 

\        , 

1. 

II 

1. 

II. 

a. 

0 

• 

1862-63 

+    '  ' 

-f  0.06 

u 

- 

1864 

o 

o 

u 

o 

1865-68 

o 

0 

u 

-  0.04 

- 

+•  0.06 

0 

-  0.04 

1870 

+  0.06 

o 

o 

-  0.04 

1871 

0 

o 

0 

o 

•87a 

0 

o 

o 

—  0.04 

1873-74 

0 

o 

o 

0 

Haying  applied  these  corrections  throughout  their  several  years,  the  ( iieenu  ieh 
and  Washington  obMrtatioiU  were  eonsidered  strictly  COmpMable;  and  when  the  moon 
\\.i-  i.l.xi  r\fd  ;it  I. nili  uliM  r\;ilorii  -  mi  the  same,  day,  the  mean  ul'  the  corrected  talmlar 
errors  \\.i-  taken.  Tin-  mean  uiitMandini:  talmlar  error  for  each  year  now  In-conic*  aa 
follow  -  : 


Year. 

Yew. 

1862 

—  2.1 

1863 

1X64 
I865 

-0.9 
+  0.4 

-f  1.4. 

-f-2.2 
-f 

+  4-' 


f,   u 


A 


1870 


-f  5-' 
+  5-6 
+  6.6 


Year. 


+  8.6 
+  9-7 


These  i|ii;intitics.  with  the  si^'ii  ehaie.'ed,  simuM  lie  considered  as  currectiuns  tu  the 

fundamental  argument,  and  we    have    tu  determine   the  eorres| diiiL'  collection  to   the 

ii^lil    a»ceiisions  which   are   to    !•••    applied    tu  the  indi\idiial  talmlar  rrn-r-.      Tu 
them  to  coiifdioiis  of  true  lunirilude,  they  an-  to  lie  multiplied  liy  the  factor 

I  -f-  2  I'  CO*  g  =.  I  -f  O.  I  1  CO«  g 


14 


Tfog  corresponding  factor  for  Correction  of  riglit  ascension  is,  with  sufficient  approx- 

imation, 

da—  (i  -f-o.  i  i  cos  g  —  0.04  cos  (2  /  —  6)  —  0.09  cos  2  /)  6  A 

III  tlii.s  formula,  eSA  represents  tlit-  correction  to  the  mean  longitude,  while  we  may 
suppose  /  to  represent  indifferently  the  mean  or  the  true  longitude;  and,  during  a  period 
of  several  months  at  a  time,  we  may  represent  the  longitude  as  a  function  of  g.  The 
value  of  da  has  been  reduced  to  a  table  of  double  entry  as  a  fund  ion  of  g  and  of  tin- 
time.  To  express  the  mean  longitude;  as  a  function  of  g,  we  have 

I-       g  +        7T 


where 


2  I  =.  2  g  -}-  2  7t 

\\j  the  substitution  of  these  values,  the  expression  for  80.  becomes 
8a  —  (  j  -j-  o.  1  1  cos  g  +  A  cos  2  g  -f-  B  sin  2  g)  6\ 


A  —  —  .04  COS  (2  7T  —  9)  —  .09  COS  2  7T 

B  —      .04  sin  (2  TT  —  0)  +  .09  sill  2  7t 
The  values  of  /T,  d,  A,  and  B  for  periods  of  six  mouths  are  as  follow 


Year. 

ir 

0 

A 

B 

Year. 

rr 

a 

A 

a 

1862.0 

o 

228 

274 

+  .05 

+  .09 

1869.0 

153 

0 

"39 

.01 

.06 

1862.5 

248 

264 

4-  .oq 

+  .09 

1869.5 

'73 

129 

.07 

-  .05 

1863.0 

269 

255 

-1-  .08 

—  .04 

1870.0 

'94 

119 

-  .08 

.00 

1863.5 

289 

245 

4-  .03 

-  .08 

1870.5 

214 

110 

-   .06 

+  .05 

1864.0 

309 

235 

—   .02 

-  -07 

1871.0 

234 

100 

—  .01 

4-   .09 

1864.5 

330 

226 

-   -05 

-  -°4 

1871.5 

255 

90 

4-   .06 

4-   .08 

1865.0 

350 

216 

-   .06 

.00 

1872.0 

275 

81 

4-  .10 

4-  .02 

1865.5 

310 

206 

-   -05 

4-  .03 

1872.5 

295 

7i 

4-  .09 

-  .06 

1866.0 

31 

>97 

—   .01 

4-  .08 

1873.0 

316 

61 

4-  .04 

—   .11 

1866.5 

51 

187 

4-   .02 

4-  .05 

1873.5 

336 

52 

-  .05 

.  II 

1867.0 

71 

177 

4-  .05 

4-  .03 

1874.0 

356 

42 

—   .12 

.04 

1867.5 

92 

168 

4-  .05 

.00 

1874-5 

«7 

32 

.12 

4-  .05 

1868.0 

112 

158 

+  .04 

—   .02 

1875.0 

37 

23 

.04 

4-  .12 

1868.5 

133 

148 

+  .03 

-   .05 

The  coellicient  i -f  o.  i  i  cos  i- -}- // cos  2  g  +  .Bsin  2»- is  next  tabulated  for  each 
of  these  sets  of  values  uf  .  /  and  />'  for  every  IO°  of  g,  and  multiplied  by  the  corre- 
sponding value  of  <5A.  As  these  tables  are  superseded  by  those  given  at  the  close  of 
this  paper,  it  is  not  necessary  to  print  them. 

The  rorrecticms  of  short    period,  which  have  been  actually  applied,  are 

// 

+  0.96  sin  D 

-o.i;,  sin  (D  +  g1) 
-f-  0.09  sin  x' 

-  0.62  sin  (2  g  —  4  g'  -f  2  u>  —  4  a/) 


15 


Tin-  lii>t   llin  i-  li.iM-  IM-I-II  (  omliinril  lulu  a  -iii^l'-  "in    n|'  i|inilil<>  ai  u'limi  nl .  in  \\  liirli 
tlic  ari!iiiiirnN  an-  l>  ainl  tin-  month;   (In-  latter  C(irrcs|MHi(lilin  t.i  i.'  .      The  li-mis  (|r|».-nil 
•  tit  on   this   arumnriit    .>"     -"    -mall    lli.it    they  m.i\    In-    i  .--.n  ii.-.l    .1-    rmislant    during  nil 
rntirr  liinlitll. 

Ill  tlii^  •..inn-  t.iUi-  i>  inrlinlril  ;i  partially   ronjrrlural  nm  rrtioii  tiir  tilt- variations  ol 

tin-   n I's    si-mi-iliamrtri.      Tin-    ronvctiou    to    Haulm's    \aln<-    lias    lu-cn    assumed    M 

—  2".o,  \\lirn  tin-  moon  is  in  tin-  m-inhliorli I  of  the  sun,  >..  that  IHT  linili  is  MTV  I'ainl. 

and  us — o"  ..\  alter  the  close  nf  evenim;  t\\ilii.'hl.  I  let  \\een  t  \vii  hours  ol  '  rlonuat  ion 
ami  tin-  rlox,-  ,,|"  t  u  iliiiht.  it  is  a.-Mimed  to  increase  uniformly.  Tin-  .sum  of  tin-si-  four 
correction-.  i>  nixeii  in  the  lollowini:  talilc  : 


i  iK-r  I.IMII. 

|| 

V  e  c 

Jj  c  c 

Q 

Ian. 

Mar. 

Apiil. 

May. 

lunr. 

July. 

Aug. 

Oct 

Nov. 

P.. 

|H 

'• 

V 

M 

—    ii 

X  •    « 

—    11 

—       12 

1  1 

4-17 

+      17 

+  i  8 

-       9 

+  1.8 

* 

f  1-3 

-r  1.3 

+  S.I 

+     l.l 

+  l.o 

t    i.-i 

+  1.8 

+  1.6 

*•     I       J 

111 

4.    i    c 

-       Q 

g 

-       7 

+  •  5 

+  15 

+  1.8 

+  1.8 

+  1.8 

+  1.8 

H.6 

+  1-4 

+  1.4 

>r  1.5 

-       7 

-       6 

+  1.5 

+  •   4 

+  1.4 

+  1-5 

+  1.6 

+  »  5 

+  1-5 

+   1-4 

+  l.l 

+  1-3 

-f   1.4 

+  1.4 

-       6 

-       5 

+  1.4 

+  1-3 

+  i  3 

+   l.l 

+  1-4 

+  1.4 

+  '.3 

+  I.I 

+  1.1 

+    1.8 

+  «  3 

+  1.4 

-       5 

—        4 

4   1  o 

4-   1    1 

»   I   > 

—            t 

—        1 

+  0.9 

+  0.8 

+  0.8 

+  0.8 

+  0.7 

4-  0.7 

+  0.8 

+  o.S 

+  0.8 

+  0.8 

+  0.9 

+  0.9 

—         2 

1 

•f  0.6 

+  0.6 

+  0.6 

+  0.6 

t  0.6 

f  0.6 

+  0.6 

I 

o 

•  s 

. 

\\ 

II 

'   =  = 

F.COND  I.I  MM. 

t: 

Jan. 

Feb. 

Mar. 

A,,  i,l 

May. 

June. 

July. 

Aug. 

Oct. 

Nor. 

Dec. 

.. 

M 

„ 

•  • 

0 

-0.4 

-0.4 

-0.4 

-0.4 

-0.4 

-0.4 

-0.4 

-0.4 

-0.4 

-0.4 

-  "-4 

-0.4 

0 

+     1 

-0.6 

-0.6 

-0.6 

-0.6 

-0.6 

-0.6 

-0.6 

-0.6 

-0.6 

-0.6 

-0.6 

-0.6 

+       1 

+      s 

-0.8 

-0.7 

-0.8 

-0.8 

-0.8 

-0.8 

-0.7 

-0.7 

-0.8 

-0.8 

-0.8 

-0.8 

+     t 

+      3 

—  1.1 

-0.9 

—  l.o 

—  l.o 

-0.9 

-0.9 

-0.9 

-0.9 

-  l.o 

-  l.o 

—  1.0 

—  1.1 

+       3 

+       4 

—  i.l 

-  i.l 

—  1.1 

—  i.l 

—  i.l 

—  1.0 

—  l.o 

i  .  i 

—  l.l 

-  I.l 

-  i.l 

-  l.S 

4-       4 

+       5 

-  1-4 

-  i.l 

-  l.S 

—  1.1 

-  «-4 

'    » 

-  1.4 

—  l.S 

-  »-3 

-1  3 

-  1-4 

-1.4 

•        - 

+       6 

-  1.4 

-1.3 

-  «-3 

-  1.4 

-  «-5 

-  '-5 

-  1.5 

-»-5 

-  1-4 

-  «  4 

-  1-4 

-'•5 

+      6 

+       7 

-  «-S 

-  «-3 

-1.6 

-1.6 

-1.7 

-  «-7 

-1.8 

-  1-7 

-1.6 

-  «-5 

-  «-5 

-  »-S 

+       7 

+       8 

-  1.4 

-1.5 

-1.8 

-1.8 

-1.9 

-1.9 

—  l.o 

-1.9 

-  1-9 

-•-7 

-  1-5 

-  «-5 

•        ' 

+      9 

-1-7 

-1-7 

—  l.o 

-  «-9 

-l.o 

—  1.0 

—  S.I 

—  l.l 

—  l.l 

-1.8 

-1.8 

-1-7 

+     10 

-1.9 

-  1.9 

-  1.0 

—  l.o 

—  S.o 

—  1.1 

-  J.I 

—  8.8 

-*-3 

—  l.l 

—  l.o 

-1.9 

+     n 

-1.0 

-1-9 

—  i.l 

—  l.o 

—  S.o 

—  8.1 

-  1.1 

-8.3 

-8.4 

-i.l 

—  l.l 

-  8.0 

+     n 

+     11 

—  1.1 

-8.0 

—  S.I 

—  l.o 

-  l.l 

—  1.1 

-1-3 

-8.4 

-  »-4 

-8.4 

-8.3 

—  8.1 

+      IS 

+     "3 

-  1.1 

-  i.l 

—  1.1 

—  8.0 

—  1.1 

—  1.1 

-8-3 

-a  4 

-8.5 

-8.5 

-8.4 

-8.3 

+      14 

-..3 

—  l.l 

—  S.I 

—  S.I 

—  l.l 

-a-3 

-8.4 

-8.5 

-8.6 

-8.5 

-«.s 

-1.4 

•    I. 

16 

By  the  application  of  the  foregoing  corrections  to  tin:  errors  of  the  moon's  tabular 
right  ascension,  these  errors  may  be  supposed  to  be  reduced  to  very  small  quantities, 
depending  on  the  errors  of  the  lunar  elements,  with  which  they  are  connected  by  the 
equation 

dot   „ ,  da.  „  „          da.  „ . 

Sa  —  ~—  61  +      -60  +  _  Si, 

til  d9  di 

the  differential  coefficients  having  the  values  given  on  page  12.      When  we  substitute 
these  values,  the  expression  for  da  will  contain  the  terms 

(+.018  SO  —  .037  So)  cos  (2  1  —  0} 

—  .087  da  cos  2  / 
+  .018  69  cos  0 
+  0.21  Si  sin  9 

—  0.2 1  Si  sin  (2  /  —  6) 

If  we  represent  the  sum  of  these  terms  by  P,  we  shall  have 

SI  =  Sa  —  P 

In  the  investigation  of  the  corrections  to  the  moon's  eccentricity  and  longitude  of 
perigee,  the  terms  of  P  maybe  entirely  neglected.  This  arises  from  the  circumstances 
that  the  appreciable  terms  of  I  or  a  arising  from  the  errors  of  these  elements  have  the 
same  period  with  g,  the  mean  anomaly,  while  P  contains  no  appreciable  periodic  term 
depending  on  g.  The  outstanding  portion  of  Sa  probably  averages  not  more  than  one 
second  or  two  at  the  utmost,  so  that  the  term  .037  Sa  is  quite  insignificant.  The  term 
.018  SO  may  have  a  constant  value  of  o".25,  more  or  less;*  but  the  short  period  of  the 
term  2  I — 9,  and  its  incommensurability  with  the  period  of '  g,  permit  of  this  error 
being  regarded  as  fortuitous.  The  same  remark  applies  to  the  terms  .087  Sa  cos  2  I 
and  0. 2 1  di  a\\\  (2  I — 0).  The  only  remaining  terms  have  the  period  of  0,  which  is 
more  than  eighteen  years.  The  effect  of  these  possible  errors  is  then-lore  eliminated 
in  the  mean  correction  for  each  year,  which  has  been  already  applied  to  the  errors. 

To  determine  the  correction  to  the  eccentricity  and  longitude  of  the  perigee  result- 
ing from  each  year's  observations,  the  residuals  in  right  ascension,  after  the  application 
of  the  three  corrections  already  described,  have  been  arranged  according  lo  the  values 
of  the  mean  anomaly  to  which  they  correspond.  The  results  are  shown  in  the  follow- 
ing table,  which  gives  for  certain  limits  of  mean  anomaly  in  the  first  column,  firstly,  the 
sum  of  the  residuals  (tabular  minus  observed)  in  right  ascension,  corresponding  to  all 
the  values  of  mean  anomaly  between  those  limits;  and,  secondly,  the  number  of .  the 
residuals.  In  taking  these  sums,  the  observations  at  the  two  observatories  are  counted 
separately,  so  that  when  observations  were  made  at  both  observatories  on  the  same 
date,  the  sum  of  the  residuals  is  taken,  and  the  observations  count  2  in  the  column  N. 

*  It  i»  afterward  found  that,  the  vulno  of  tliift  product  is  only  o".o8. 


17 


Sums  of  errors  of  moon's  corrected  r'niht  <ucaut<"  -•«/»//«>•  <//  ti rerun  «  > 

hinqton. 


IS6* 

1863 

1864 

Limits  of  mean 

anomaly. 

I.U 

H€ 

\ 

N. 

Urn 

N. 

•            • 

0   10      10 

+     3-9 

4 

*   21.5 

10 

+   ' 

9 

•«•       1-4 

7 

10   10      *0 

-1-     3-6 

6 

+    ' 

12 

+ 

7 

* 

4 

•o  to    30 

—      0.2 

$ 

+    14    2 

8 

+     S-8 

S 

-      0.3 

10 

JO    10      <0 

+     9-3 

8 

+  23.7 

II 

4-5 

7 

-    o.$ 

S 

40  lo     5° 
50  lo    60 

+     0.3 

8 

+     9-8 

9 

-      1.6 

10 

—      I.I 

6 

60  to    70 

+     8.9 

10 

-     4-3 

7 

+ 

5 

-     6.1 

7 

70  to    So 

-     3-7 

4 

+     7-0 

10 

-     7.0 

6 

80  lo    99 

+     6.7 

7 

-     6-7 

-     11.2 

9 

- 

90  to  too 

+     3-9 

6 

-     3-3 

9 

-       3-4 

6 

-     8.$ 

7 

too  to  no 

+     3-9 

it 

-    0.4 

5 

-       J.I 

S 

-     0.7 

S 

1  IO   to   I2O 

1*0  to  130 

-     3-» 

8 

-     3-9 

7 

+       O.I 

5 

-     5-5 

6 

130  to  140 

•+•           II 

140  lo  15^ 
150  to  160 

•  —    o.i 

S 

-     18.2 

-  • 

7 

+     >  $ 

4 

160  to  170 

-    8.8 

4 

-   19-7 

6 

+     2.5 

6 

+     4.3 

5 

170  to  180 

-     $-7 

4 

-    9-9 

7 

-     S3 

5 

+ 

6 

180  to  190 

-   17-4 

9 

-  33  « 

14 

-     8.6 

7 

-f     8.9 

• 

100   10   300 

-  15  5 

7 

-       -3 

4 

-     0.6 

4 

+  13-2 

8 

too  lo  210 

-     3-8 

10 

—          .0 

6 

-     6.4 

9 

+     7-8 

8 

210   to  MO 

-      0.2 

2 

-       -9 

9 

-     2.9 

8 

+  «3  « 

7 

MO    10   930 

-  28.9 

. 

-       -5 

10 

+     3.'. 

7 

+     <   i 

S 

130  to  240 

-   7.3 

7 

-         9 

7 

+     0.8 

7 

+   10.3 

S 

140  to  250 
150  10  260 

—    s.o 

4 

+     7-6 

8 

+  11.5 

8 

+     7-3 

j.    16  • 

7 

12 

too  to  27° 
270  to  280 

37 

5 

+  11.3 

9 

+  25.3 

ti 

r     •**.  * 

+     7-6 

It 

- 

+     4-7 

7 

t-     0.8 

S 

rt.i 

- 

+     9-6 

- 

200   10   300 

-     «-3 

i 

+  «5-9 

- 

+     6.6 

4 

+     $-8 

II 

300  to  310 

+     3-o 

J 

+  23.5 

9 

+     7-8 

6 

+    10.  1 

7 

310  lo  320 

+     2.3 

2 

+    22.6 

6 

+     6.4 

S 

-t-    16.4 

10 

320  lo  330 

-      2.8 

: 

-t-  18.2 

9 

+    i 

7 

-f-    14-5 

7 

330  to  MO 

+     9-5 

6 

+      1.2 

7 

+   18.5 

10 

+   l6.T 

u 

340  to  350 

-f-   11.8 

- 

+      7-2 

- 

+      4   * 

7 

+     7-6 

7 

350  to  360 

•+•   13.6 

S 

+    14-4 

8 

+    16.5 

6 

•*•     S-J 

9 

+  106.4 

22$ 

-t-  222.0 

287 

+  I8T-I 

236 

+  205.9 

255 

—  116.0 

-144-7 

-  71.0 

-  46.6 

-     9.6 

+    78.J 

+  «S9-J 

3M 


18 


Sums  of  errors  of  moon's  corrected  right  ascension,  Sfc. — Continued. 


1866. 

1867. 

1868. 

1869. 

Limits  of  mean 

anomaly. 

£<Ja 

N. 

Ida 

N. 

£<Ja 

N. 

Z(!a 

N. 

o  to     10 

-     1-7 

6 

II 

+     7-4 

5 

4-2 

4 

-   10.7 

4 

10  to     20 

-     2.5 

4 

-     5-0 

2 

+     3-9 

7 

—     4-2 

4 

20  to     30 

-     7-5 

3 

1-7 

4 

-     2-5 

3 

-     0.8 

6 

30  to     40 

-     7-1 

5 

-     7-5 

3 

9-4 

6 

+      4-2 

5 

40  to     50 

-  14-5 

7 

+     5-5 

4 

-     9.0 

5 

+    II.  O 

6 

50  to     60 

-     0.7 

I 

—      2.O 

4 

—     0.7 

7 

+     5-5 

3 

60  to     70 

+     1-3 

5 

-     8.5 

4 

+      2.2 

7 

+     3-1 

5 

70  to     80 

+     5-3 

6 

•     4.8 

3 

+     4-1 

8 

+     7-7 

7 

80  to     go 

+     1.6 

6 

-     3.6 

I 

+    12.2 

7 

+     8.0 

8 

90  to  too 

+     3-9 

4 

+      2.6 

5 

-     0.3 

4 

+  16.8 

8 

100  to  no 

+     4-4 

9 

—     0.6 

5 

+    14.9 

7 

+     5-1 

9 

no  to  120 

+     4-» 

8 

+     3-9 

5 

+      9-8 

6 

+     8.3 

6 

I2O   to    130 

5.4 

8 

+     1.6 

7 

+     4-1 

5 

+  14.5 

7 

130  to  140 

+     3-4 

6 

+     4.1 

6 

+    10.2 

8 

+     7-5 

8 

140  to  150 

+    10.  I 

9 

+     1.9 

7 

+     5-2' 

7 

+     3-1 

6 

150  to  160 

-     4-1 

6 

-      2.6 

7 

+      2.1 

9 

+  20.3 

7 

160  to  170 

+     3-3 

7 

+     6.8 

5 

+     1-3 

8 

+     3-7 

3 

170  to  180 

—      O.I 

7 

-     5-0 

8 

+     0.8 

7 

+    12.2 

7 

180  to  190 

+     0.8 

6 

-     0.3 

2 

+  12.3 

8 

+      7-0 

5 

190  to  200 

+     5-9 

6 

+      2.0 

4 

+  17-9 

6 

+      6.3 

4 

200  to  210 

-     3-2 

6 

+      2.8 

6 

+     5-2 

5 

+    10.  I 

5 

210  to    220 

+     0.3 

6 

-     1-7 

4 

+  13.0 

8 

+    12.2 

5 

22O   tO   230 

-     5-4 

4 

+  12.9 

9 

+     4-8 

4 

+    12.3 

7 

230  to  240 

+     4-1 

8 

-f-      8.2 

6 

+  15-2 

9 

-     1-3 

3 

240  to  250 

-     1.8 

7 

+  25-4 

9 

+     7-4 

8 

-     6.4 

6 

250  to  260 

+     9-4 

7 

+     0.9 

3 

+  14-2 

8 

-     3-6 

2 

260  to  270 

+      2-7 

7 

+   11.7 

6 

-     5-0 

2 

-  17-3 

7 

270  to  280 

+     9-7 

4 

+     3-3 

4 

+       I.O 

7 

—  18.8 

5 

280  to  290 

+  n.  6 

12 

+     7.0 

7 

-     9-i 

5 

-  21.4 

6 

290  to  300 

+     4.0 

4 

+    0.7 

3 

-     3-2 

8 

—  13.6 

3 

300  to  310 

+     6  7 

4 

+  16.5 

7 

-     8.0 

2 

-     4-8 

2 

310  to  320 

+     3-4 

2 

+     2.3 

5 

-  13..  8 

8 

-     0.8 

t 

320  to  330 

+     7-7 

5 

+      0.2 

5 

—   10.6 

9 

-     4.2 

2 

330  to  340 

+     9.1 

5 

+      3-5 

6 

-    II-  7 

6 

-   18.5 

6 

340  to  350 

+  10.8 

6 

5-4 

7 

-     9-8 

5 

—  10.6 

4 

350  to  360 

+    92 

7 

-     7-2 

4 

-  18.3 

6 

—      2.2 

5 

+  132-9 

213 

+131.2 

182 

+  161.8 

229 

+  178.9 

187 

-   54-0 

-  55-9 

—  115.6 

-139-2 

+  78.9 

+  75-3 

+  46.2 

+  39-7 

IS* 


SHMU  of  error*  of  tuoon't  corrected  right  ascension.  Jp. — Concluded. 


Umiuofmean 

1870 

1871 

1871 

1 

»87> 

1874 

anomaly. 

Zrf« 

N. 

Z4* 

N. 

X<U 

N. 

ZJa 

N. 

Irfa 

N. 

•           • 
OIO     IO 

-     7.» 

S 

H 

-     3  • 

5 

• 

4-     6.5 

6 

-     4-3 

6 

4- 

4 

1010     20 

-      2.2 

S 

4-     1.7 

n 

4-     8.5 

IO 

4-     S.» 

4 

t"     5-9 

5 

•010     JO 

4-      $•« 

6 

-     0.3 

7 

4-      5-5 

8 

4-     S-» 

8 

4-   12.5 

6 

jo  to   40 

4-  10.7 

8 

4-     6.4 

7 

4-   II.  8 

7 

4-      3-4 

3 

4-     5.1 

| 

4010    jo 

+    It.  2 

8 

4-  16.7 

9 

4-     6.0 

4 

4-     6.6 

4 

4-      4-4 

5 

so  to   to 

-     7-» 

S 

4-     9-7 

6 

4-   13.2 

6 

4-     4-1 

7 

4-    a.i 

S 

to  10   70 

+      1.0 

9 

+   18.9 

8 

4-    10.4 

3 

4-   13-4 

6 

4-    I 

4 

7010   to 
to  to   oo 

-     2.6 
4-   12.  0 

5 

it 

4-    10.2 
4-117 

7 

4-    12.4 

8 

4-   13-5 

3 

4-     6.6 

6 

T^ 
go  lo  too 

+   IO.I 

8 

•  *  •  / 

4-  12.5 

3 

4-     9-8 

4 

4-     5-1 

a 

4-     5-9 

7 

too  lo  no 

4-  10.8 

4 

4-    19-7 

8 

4-    13-0 

6 

4-     1.5 

3 

4-  10.9 

6 

no  to  120 

4-     5-8 

6 

4-     8.1 

4 

4-   18.7 

6 

4-       .3 

a 

4-     4.6 

4 

1  JO  IO  I  JO 

4-    IO.I 

7 

4-     9-7 

5 

4-    18.3 

7 

4-        .1 

5 

-     4.7 

6 

13010  140 

4-    IO.I 

S 

4-  15.4 

5 

4-     0.2 

a 

4-        -3 

3 

4-     l.B 

3 

140  10  150 

+  18.2 

8 

4-      2.1 

3 

4-      2.9 

3 

4-       .4 

5 

-    0.8 

7 

150  to  160 

4-    4-4 

3 

4-     3-0 

7 

4-     2.1 

8 

-       -9 

3 

4-      1-3 

S 

ito  to  170 

+     8.8 

S 

4-     8.7 

4 

4-    6.6 

5 

—       .4 

4 

-    IO.I 

9 

17010  ito 

+    6.9 

3 

4-     6.2 

6 

—      1.2 

3 

-       -7 

3 

—       1.0 

6 

180  10  190 

+     I  8 

i 

4-     3-9 

4 

4-      1.9 

4 

—         .2 

4 

+    $.0 

6 

190  lo  too 

4-     7-5 

4 

4-     3-5 

3 

—      1.2 

5 

-       .6 

6 

—       1.0 

2 

900  10  210 

4-      2.1 

5 

+     t.o 

3 

—      2.2 

6 

-       -9 

a 

4-     3-7 

• 

aio  lo  no 

-     *-3 

2 

-     2.6 

a 

4-     i.  a 

3 

-       .6 

3 

-     5-0 

5 

MO  10  330 

-     *•$ 

3 

-     9-3 

7 

-    7-  a 

5 

—      O.I 

4 

—  16.0 

7 

230  to  940 

-     0.4 

S 

-     3-» 

6 

-     4-8 

5 

-     35 

1 

-  «3-5 

4 

140  to  250 

-     9-7 

5 

-    9-« 

8 

-    6.5 

3 

-     7-5 

5 

-  «5-l 

8 

250  to  260 

—  la.i 

6 

-     5  » 

5 

-     9  « 

4 

-     7-1 

4 

-  33.0 

5 

ato  to  770 

-     »-3 

2 

-     4-6 

5 

-  13.8 

8 

-     8.6 

3 

-    22.6 

4 

27010  ato 

-  12.9 

8 

-     7-1 

7 

8.4 

5 

-    4-3 

4 

-  15.6 

4 

ato  to  290 

-     S-6 

3 

-     *-7 

6 

-  16.7 

9 

-  10.8 

6 

-    9.1 

3 

29010  300 

-     $-S 

4 

4-     4-0 

4 

-  10.3 

8 

-    9.8 

4 

-  13-4 

8 

300  to  310 

-     4.0 

4 

-     9-5 

6 

-    9-5 

5 

-     1.8 

1 

-     9-' 

9 

310  to  320 

-     8.7 

3 

-    6.6 

S 

-     5-6 

4 

-     3  » 

4 

-     5-3 

6 

320  to  330 

-  «3-5 

6 

-    4-9 

7 

-     8.5 

5 

-  ".3 

7 

-      1.4 

7 

33010340 

-     9-7 

4 

-    2.8 

7 

-     8.5 

5 

-    o.j 

8 

-     4-3 

3 

340  to  350 

-     3-6 

3 

-     1-7 

4 

-     5  « 

5 

-    9.2 

6 

4-    a.  a 

n 

350  to  360 

-     8.7 

5 

4-     6.3 

4 

4-     O.I 

6 

-     4.0 

5 

4-     2.5 

6 

+  136.6 

185 

4-179-5 

*>3 

4-160.4 

»95 

4-  96.9 

•55 

4-  95.2 

200 

-120.6 

-  71.8 

-II8.6 

-113  I 

-171.0 

+  16.0 

4-106.7 

+  41.  » 

-  16.2 

-  75-8 

Neglecting  all  terms  multiplied  liy  the  eccentricity  in    tin-  c.»-mYinits.  nidi  : 
mil  gives  an  equation  of  the  form 

Jl  -f-  2  sin  g  dp  —  2  cos'^  e  dir  •=.  r 


20 

or,  putting 

It  zz  2  Jde  —  —  2  Se 

k  —  —  2  .Je  6V  —  2  e  (Szr 
the  equation  will  be 

41  -\-  h  sin  g  +  A  cos  g  —  r, 

-Je  and  dn  being  the  errors  of  the  tabular  eccentricity  and  longitude  of  the  perigee, 
while  Se  and  STT  represent  the  corresponding  corrections. 

The  equations  are  now  solved  as  if  all  the  residuals  within  each  pair  of  20°  limits 
corresponded  to  the  mean  of  the  limit,  —  that  is,  as  if  all  between  o°  and  20°  corre- 
sponded to  g  =  10°  ;  those  between  g  —  20°  and  g  —  40°  to  g  —  30°  ;  and  so  on.  If, 
then,  we  put 

gi  —  10°  ;  g2  =  30°,  etc.  ; 

rt,  the  sum  of  all  the  residuals  in  any  one  year  corresponding  to  g  —  gt; 

nit  the  corresponding  number  of  observations; 

s{  =  sin  gt  ; 

Ci  =  cos  gt  : 
the  normal  equations  for  determining  dl,  h,  and  k,  by  least  squares,  will  be  : 


{   ^)  Jl  +  (2  nt  s?)    h  -f  (2  nt  st  ct)k  =  2  s{  r{ 
(2  nt  Ci)  41  +  (2  n,  Si  d)  h  +  (2  n{  c,2)    k  -  2  e{  rt 

The  formation  and  solution  of  these  equations  for  each  year  give  the  following 
values  of  the  outstanding  errors  of  the  lunar  elements  for  each  year: 

//  /' 

1862,  /*  —  -f  0.04  £  =  +1.23 

1863,  -0.64  +1-78 

1864,  —  1.07  +  1.09 

1865,  -1.03  -0.15 

1866,  -0.47  +0.10 

1867,  —0.93  —0.36 

1868,  +0.34  -  1.46 

1869,  +  1.67,  —  1.56 

1870,  -j-  1.48  -  1.14 

1871,  +1.65  —0.36 

1872,  +2-'5  —  O.I  2 

1873,  +  I-91  +0.16 

1874,  -(-  1.92  +0.60 

The  periodic  character  of  thcsi;  residuals  is  very  remarkable,  indicating,  as  it  does, 
either  a  hitherto  unknown  inequality  of  the  moon's  mean  longitude,  having  nearly  the 
same  period  with  the  orbital  revolution;  or  one  of  <  he  eccentricity  and  longitude  of 
perigee,  having  a  period  of  between  fifteen  and  twenty  years.  To  investigate  this  in- 
equality, we  shall  assume  that  each  value  of  h  is  of  the  form 

k  —  a  sin  (JLI  +  nt) 
and  each  value  of  k  of  the  form 

k+  a'cos(yu'  +M'/), 


L'l 

//.  ft,  .  n  .  n,  ;ui<l  //    IM-UIU'  unknown  quantitie>  i..  )»•  determined,  and  /  the  time 

in  \.-;u>  tiiiin  an\  assumed  i-|iu<-ii.  \\  <  shall  lake  for  the  eporli  tin-  inidillr  of  I  he 
period  through  whieh  the  obsenations  extend:  that  is,  1868.5.  IT.  then,  \M-  represent  the 
thirteen  values  of  //  and  k  in  chronologic-ill  order  liy  A_,,  A_,,  .....  A,,  *_„*_*  ..... 
£„  tin-  equations  of  condition  lor  It  and  k  respectively  may  In-  put  into  tin-  form 

A,  zz  A  —  a  sin  //  •  »-  /  n  —  a  cos  //  sin  •  n 
kt  rz  k  +  a'  cos  ft  r..s  /  n  —  a'  sin  //  sin  »  //. 

irdiiiif  /«,  Ar,  a  sin  ^,  or  cos  /^,  a'  sin  //,  and  or'  co«  /<  as  tin-  unknown  iiuuiititim, 
the  normal  r<|iiatioiiK  tor  ilvtiTiiiiiiing  tlicso  quantities  an-  : 

(1)  From  the  ra/uft  of  /«,. 

13  A  —  (if  co«  i  w)  a  sin  ;«    zz       —A, 
—  (i'cosiii)  A  -f-  (i'cos*  i  M)  a  sin^  =  —  -//,  C.IH  i  « 
(i'  sin*  i  M)  a  ros  //  zz  —  —  //,  >in  /  H 

(2)  from  the  rulmx  <i(  A,. 

13  k  -f  (i'oos  i  »)  a'  cos  n'  =      i'X-j 
(i*  cos  i  »)  A  -f-  (2?  cos*  i  »)  a7  cos  ^'  =      i'  k,  n  .s  i  n 
I?  (sin*  i  M)      a'  sin  n'  •=.  —  2  *,  sin  /  n 

It  will  l>e  observed  that  all  tin-  roi-trn-i«-nts  havini;  as  n  liu-tor  either  2  sin  i  n  or 
2  Kin  IN  C08  i  n  vanish. 

The  value  of  «  apparently  i«  not  readily  determined  directly  liy  least  squares:  we 
shall  therefore  a^ume  sexeral  \alnes  of  this  quantity,  and  iLM-ertain  liy  which  value  the 
conditions  can  l>e»t  be  Hfttintied.  The  following  are  the  abbreviated  values  of  the  purely 
trigonometric  summations  : 

„  Mil  6J  H 

_  i  .»  /  II  —  =Z  C 

MM  1  n 

„  13  sin  n  -|-  sin  13  M 

i  cos*  »  n  zz    °  =  c, 

2  sin  « 


2  sin  a 
K  we  solve  the  preceding  equation*,  and  put,  lor  brevity. 

C  -         C 
- 


13  c,  —  «* 


the  resulting  expressions  for  the  unknown  quantities  are: 

A  =        ''.-A*—   CEfAjCtti  in 
-in  ^  =        r^  A.  —  C'^'AjCo*  in 

a  cos  ^  zz  —  —  2"  A,  sin  »» 
*i 

A  zz         CiSkt  —  C2f*,cos  i  » 


a'sin  X  =  -- 


22 


The  period  of  h  and  k  lies  probably  between  fifteen  and  twenty  years,  which  would 
make  the  value  of  n,  or  the  annual  motion  of  the  inequality,  lie  between  18°  and  24°. 
The  following  are  the  values  of  the  various  quantities  depending  on  n  for  the  different 
values  of  n  between  these  limits : 


n 

logc 

log  a 

log  st 

logC 

logC 

log  Ca 

18 

0.756 

0.715 

0.893 

9.213 

9.172 

9-571 

1  9 

0.705 

0.707 

0.898 

9.097 

9.099 

9.506 

20 

0.644 

0.705 

0.900 

8.977 

9.038 

9-447 

21 

0.577 

0.709 

0.897 

8.858 

8.990 

9-395 

22 

0.498 

0.718 

0.891 

8.734 

8.954 

9-350 

23 

o  406 

0.731 

0.882 

8.604 

8.929 

9.312 

24 

0.291 

0.747 

0.870 

8.453 

8.909 

9.276 

25 

0.143 

0.765 

0.856 

8.275 

8.897 

9.246 

n 

2  hi  sin  in 

2  hi  cos  i  n 

£/t(  sin  in 

2/6(  cos  in 

0 

IS 

+   11.48 

+     1.96 

-     4.66 

4.66 

'9 

+  n.66 

+      1.52 

4.68 

5-04 

20 

+   11.78 

4-      1.09 

-     4-(>9 

5-40 

21 

+  11.83 

+     o  68 

4.68 

-     5-73 

22 

+   II.  Si 

+     0.29 

4.66 

6.04 

23 

+   U-73 

—     0.08 

4.62 

-     6-33 

24 

+   11.58 

0.44 

4-57 

-     6.60 

25 

t-  H-37 

—     0.78 

4-50 

-     6.86 

The  preceding  equations  now  give  the  following  separate  values  of  the  unknown 
quantities,  corresponding  to  the  various  assumed  values  of  n: 


n 

A 

a 

/< 

k 

a' 

f' 

0 

18 

O.72 

n 
'•53 

164.0 

o.73 

1.81 

160.8 

'9 

0.69 

1-53 

165.2 

0.61 

1.71 

'59-7 

20 

0.66 

i-53 

166.3 

0.49 

1.62 

158-5 

21 

0.63 

1-54 

167.2 

0.39 

1-53 

157-2 

22 

0.61 

«-55 

168.1 

0.31 

1.47 

156.0 

23 

0.60 

'-57 

169.0 

0.23 

1.42 

'54-8 

24 

0.58 

1-59 

169.8 

0.17 

1-39 

'53-6 

25 

0.56 

1.61 

170.4 

O.II 

1.36 

152.6 

There  can  be  little  serious  doubt  that  in  the  case  of  the  present  inequality  the 
theoretical  values  of  //  and  n'  should  oe  the  same;  and  it  is  also  probable  that  those  of 
a.  and  a!  may  be  substantially  identical.  The  small  differences  between  the  values  of  a 
and  a'  and  of  /*  and  //'  add  so  much  weight  to  this  probability  that  we  shall  make 


another  solution  o!'  tin-  equations  on  ill.-  Miii|>..Mtion  I  hat  a   —  a  anil  /*'  =  p.     The  nor- 
mal ci|iiations  then  lu-co : 

13  A —   ca  sin  p  •=.      2/i, 

—  ch+  1 3  nr  sin  ;<  =  —  2?A< cos  in  —  2k{  sin  i n  =  5, 
13*+  coco*  n  —      i'A, 

rA:-f  13  orcos// =      2  kicwin  —  2his\nin  =  St 
The  solution  of  these  equations  is: 


,  ___ 

'3  —  ^  "3  —  <• 


"'3*-?  if--* 

A  coni|Hirison  of  the  separate  solutions  ol  the  equations  in  A  and  k  shows  that  the 
value  of  n  which  best  satisfies  the  conditions  lies  between  .22°  and  25°.  The  values 
of  A,  k,  a,  and  ft  were  therefore  derived  only  from  the  lost  equations  for  the  last  four 
values  of  M.  For  each  of  these  separate  values  of  n,  the  corres|M>ndtng  values  of  A<  and 
4j  were  computed  from  the  formula* 

A(  rr  A  —  a  sin  (ft  -\-  i  n) 
kf  ==  k  -f-  «  cos  (ft  -j-  i  n) 

in  which,  it  will  be  remembered,  the  index  /   is  simply  the  number  of  the  year  from 
1868  ;   so  that  we  have, 

For  1862,    «'  =  —  6 

For  1863,    i  =  -5 

etc.,  etc. 

These  computed  values  of  A,  and  k,  were  then  compared  with  the  values  derived 
directly  from  observations,  and  given  on  |>uge  20.  and  the  sum  ol  the  squares  ot  the  out- 
standing residuals  was  taken.  The  values  of  the  unknown  quantities,  together  with 
the  sum  of  the  squares  of  the  residuals,  are  a.s  follow : 


• 

4 

i 

• 

f 

Z 

• 

JJ 

+  0.66 

+  O.J4 

•t 
'54 

• 
161.2 

•  • 

3.»oj 

*3 

+  0.63 

+  o.«7 

•  5» 

161.3 

J-iTO 

*4 

+  0.61 

+  o.*o 

1.51 

|6|.$ 

3«4« 

*$ 

+  0.58 

+  0.14 

«-49 

161.7 

J  441 

The  sum  of  the  squares  becomes  a  minimum  for  n  =  22°. 8.  showing  a  period  ol 
the  inequality  of  I57.8,  with  a  possible  error  of  a  \  e.ir  or  more.     The  formula-  for  A,  ami 

ki  thus  become : 

A,  =  +  o".64  —  i".52  sin  (i6i°.2  +  22°.8  i) 

kt  =  +  o".28  +  i".52  cos  (l6i°.2  +  22°.8  i) 


from  which  we  have  the  following  comparison  of  the  computed  and  observed  values  of 
hi  and  kt : 


Year. 

fc 

« 

C. 

0. 

O.-C. 

C. 

O. 

O.-C. 

1862 

4-   O.OI 

+  0.04 

+  0.03 

4-  1.67 

+  1.23 

•  0.44 

1863 

•  0.48 

—  0.64 

0.16 

4-  1.32 

+  1.78 

+  0.46 

1864 

-  0.79 

-  1.07 

—  0.28 

+  0.80 

+  1.09 

4-  0.29 

1865 

-  0.88 

—  1.03 

-  0.15 

+   O.22 

-  0.15 

-  0.37 

1866 

-  0.74 

-  0.47 

+  0.27 

-   0.38 

+   O.IO 

4  0.48 

1867 

-  0.37 

-  0.93 

—  0.56 

-   0.85 

—  0.36 

+  0.49 

1868 

4-  0.14 

+  0.34 

4-   O.2O 

-  1.16 

-  1.46 

—  0.30 

1869 

+  0.74 

+  1.67 

+   0.93 

-  1.23 

-  1.56 

-  °-?3 

1870 

+  1-33 

4  1.48 

+   0.15 

-  1.07 

1.  14 

—  0.07 

1871 

+  1.  80 

+  1.65 

-   0.15 

—  0.70 

—  0.36 

+  0.34 

1872 

4  2.09 

+  2.15 

4-   O.O6 

—  o.  18 

—   0.12 

+  0.06 

1873 

4-  2.J5 

+  1.91 

O.24 

+  0.42 

f-  o.  16 

—  0.26 

187; 

+   I.gS 

+  1.92 

—   O.O6 

+   I.OO 

4-  0.60 

0.40 

The  probable  residual  for  each  year  is  o".27. 

We  have  supposed  the  hypothetical  inequality  of  longitude  to  be  of  the  form 

4v  —  ^  sin  g  -f-  kt  cos  g. 

Substituting  in  this  the  periodic  part  of  ht  and  k^  and  replacing  i  by  /,  which  now  repre- 
sents the  time  in  years  from  1868.5,  it  becomes: 

or 

Jv=  i".52  sin  [g  +  22°.8  (Y-  1857.5)] 

The  entirely  unexpected  character  of  the  periodic  term  thus  brought  to  light  ren- 
ders its  verification  by  a  longer  series  of  observations  very  desirable.  For  this  purpose, 
we  need  comparisons  of  observations  previous  to  1862  with  Hansen's  tables,  because 
none  of  the  older  tables  with  which  comparisons  have  been  made  are  accurate  enough 
for  the  purpose.  Now,  the  Greenwich  Observations  for  1859  contain,  as  an  appendix,  a 
comparison  of  the  longitudes  and  latitudes  from  Hansen's  tables  with  Greenwich  observa- 
tions from  1847  to  1858  inclusive  ;  and  I  have  utilized  the  comparison  of  the  longitudes 
derived  from  meridian  observations  in  the  following  way  : 

A  list  of  limiting  dates  to  tenths  of  a  day  was  made  out,  including  the  whole  twelve 
years,  and  showing  between  what  dates  the  moon's  mean  anomaly  was  found  in  each 
sextant.  The  sum  of  the  errors  in  longitude  given  by  the  meridian  observations  was 
then  taken  during  the  period  that  the  anomaly  was  found  in  each  sextant.  None  of  the 
corrections  found  in  the  first  part  of  this  discussion  were  applied,  for  the  reason  that 
most  of  them  could  be  treated  as  accidental  errors,  and  the  means  could  be  taken  so  as 
nearly  to  eliminate  the  effects  of  the  larger  ones.  A  specimen  of  the  form  chosen  is 
here  given.  Under  each  of  the  several  values  of  g,  given  at  the  tops  of  the  several 


2f> 


columns,  is  .shown,  f,r*tly.  th,-  .lati>  at  which  -  lia.l  that  particular  value:  ami.  Moomlly, 
the  MIL  i  o!  tli.-  n-si.luals  in  longitude  .lurin-  the-  period  of  4^.6  l.elweei,  tliat  dale  mid 
this  one  next  following  together  with  the  number  of  the  residuals,  the  latter  I.ein:/  in 
small  subscript  (inures. 


,.*. 

/=*»'  + 

/=!»*+                 /  =  l8o'  + 

/  =  140'  + 

/-  **>'  + 

1847- 
Jan.     19.  D-  1.9, 

1847. 
Jan.     14.1-  3.1, 

•  847.     " 
Jan.       1.1+   i  j, 

Jan.       S.I      .  . 

1847 
Jan.     10.4+  1.9, 

1847-      " 
Jan.     15.0      .   . 

Feb.    16.1—   i.o, 

Feb.    to.  7  +  0.4, 

Jan.     18.8+  3.  i( 

Feb..     1.4+  3.7, 

Feb.      7.0+  5.6, 

Feb.    11.6      .  . 

Mar.    15.7      .         M  „     10.3-  3.0, 

Feb.    15.34-  4  S» 

Mar.      1.9+  3.7, 

Mar.      6.5+  2.3, 

Mar.    ll.l       .  . 

April  11.3       .   .    I  April  16.9      .  . 

Mar.    24  94-  6.1, 

Mar.    lu.s-  0.4, 

Apiil     3.1 

April     7.7+  3.1, 

May      9.8       .   . 

May    14.4       .    . 

April  ai.  sf   3.1, 

April  16.0+  1.7, 

April  30.6      .  . 

May      $.14-  l.o, 

June     6.2+  1.8, 

June    10.8       .   . 

May     19.0+   1.8, 

May    13.6+  3.5, 

May    18.1—  0.3, 

June      1.8+  4.1, 

July      3.8+  4.4, 

July      8.4      .    . 

June    15.4-   1.4, 

June    20.0—  0.6, 

June    14.6—  i.i. 

June    19.1+   1.9, 

Aug,     0.4-  0.3, 

Aug.     5.0      .   . 

July     13.0      .  . 

July     17.6-   1.6, 

July    11.1+  1.9, 

July     26.84-   6.8, 

Aug.    18.04-  5.9, 

Sept.     1.6+  3.8, 

Aug.     9.6      .   . 

Aug.    14.1      .  . 

Aug.    18.8—  8.4, 

Aug.    13.4+  8.1, 

Sept.   24.64-11.1, 

Sept.  19.1 

Sepl.     6.1 

Sept.    10.8       .   . 

Sept.    15.  4f   3.6, 

Sepl.    20.0+  i.o, 

Oct.     M.  14-12.14 

Oct.     16.7  f  8.7, 

Oct.      3.8      .  . 

Oct.      8.4      .  . 

Oct.     13.0       .   . 

Oct.     17.6-  7.3, 

Nov.    19.7-  1.2, 

Nov.   13.34-  9.3, 

Oct.     31.34-  l.o, 

Nov.     4.9       .    . 

Nov.     9.5 

Nov.    14.1       o.o, 

Dec.    16.1-  3.4, 

Dec.    10.7 

Nov.   17.6+11.4! 

Dec.     1.1      .  . 

Dec.     6.8      .  . 

Dec.    11.4-  1.6, 

Dec.    15.3+  0.7, 

Dec.    29.9+  0.7, 

D«e     34.$ 

Dec.    39.1-  1.8, 

1848. 

1848. 

1848. 

1848. 

1848. 

1848. 

Jan.     11.7-  7.3, 

Jan.     17.3       .    . 

Jan.     21.9       .   . 

fan.     26.  $4   o.i, 

Jan.     31.1       .  . 

Feb.      4.7       .   . 

Feb.      9.3-  8.1, 

Feb.     13.9—  6.1» 

Feb.     18.5-    1.4, 

Feb.    23.1-   1.4, 

Feb.    17.7—  0.8, 

Mar       33       .   . 

Mar.      7.9-   1.8, 

Mar.    11.5-  4.3, 

Mar.    17.1+  4.7, 

Mar.    21.7      .  . 

Mar.    16.3      .  . 

Mar.    30.9      .  . 

April     4.5       .    . 

April     9.1-  4.1, 

April  13.7-  1.5, 

April   18.3-   1.8, 

April  11.9+  1.9, 

April  27.5       .   .   i 

May      2.0       .    . 

May      6.6-  7.4. 

May     II.  if    i.o, 

May     15.84    1.4, 

May    10.4+  9.0, 

May    25.0       .   . 

May    29.  ft       .    . 

June      3.1—  8.9, 

lunc      7.8—  0.9, 

June    12.4—  o.l, 

June    17.0       .    .      June    11.6+   2   i, 

June    26.2       .    . 

June    30.8—  2.6, 

July      5.4-  0.6, 

July     10.0-  4.8, 

July     14.  6  f  to.  4, 

July     19.1-  O.I, 

July     23.7       .    . 

July    28.3       .    . 

Aug      1.9-  5.4, 

Aug.     6.$-  3.1, 

Aug.   ll.l 

Aug.    15.74-17.4, 

Aug.    20.3+   1.2, 

Aug.   14.9      .  . 

Aug.  19.5      .  . 

Sept.     3.1-  6.7, 

Sept.     7.7-  5.1. 

Sept.   11.3+IS  7i 

Sepl.    16.9422.$. 

Sepl.    ll.$ 

Sept.  16.  i 

Sepl.   30.7-   i.o, 

Orl.        $.3+   1.0, 

Oct.       9.8+  83, 

Oct.     14.4+   $.1, 

Oct.     19.0  ... 

Oct.     13.6      .  . 

Oct.     18.1       .   . 

Nov.      i.8f  1.9, 

Nov.     6.4—  4.9, 

Nov.    11.  o*   6.9, 

Nov.    i$.  6412.  ft, 

Nov.    10.1 

Nov.    14.8      .  . 

Nov.    19.4-  9.4, 

Dee.      4.0-  7.1, 

Dec.      8.$-  6.5, 

Dec.    13.1+   5.1, 

Dec.    17.  7f  7.0, 

Occ.    22.3      .  . 

Dec.    16.9       .    . 

Dee.    31.5-  $.71 

If  we  follow  any  one  of  these  vi-rtical  mluimis,  we  -Indl  timl  that  the  date*  corre 
s|Niiul  successively  to  all  points  of  tin*  lunation  in  a  |ieriod  of  412  d.i\«  The  first 
observations  of  each  period  will  be  tin-  last  i»n«-s  uf  tin-  lunation,  an- 1  tin-  la>t  on.->  tli..^,- 
made  immediately  alter  new  moon.  llHurrn  i-acli  pair  o|'  pnioiU  will  \»-  a  nap.  L'''u- 
.-tally  of  three  or  four  months,  during  which  tin-  moon  was.  at  tin-  COfTOponding  |Mimt> 
of  mean  anomaly,  too  near  the  sun  to  be  obsrm-d.  Il  tin-  oli>«-i \alions  are  equally 
scattered  through  each  period,  all  the  rrrors  arisinir  from  rrnirx-ous  -i-iiii-diaiii«-t)-r  and 
parallactic  inequality  will  l»c  eliminated.  The  umrral  minuteness  of  tln-sr  rrmrs.  and 
their  approach  to  a  lialance  durinc  each  ol  tin-  pi-ritnU  in  ijiu-tion.  an-  such  as  lo  render 
them  insiunilicnnt,  if  we  take  the  mean  roults,  not  l»y  years,  but  l>y  |ierio.l-  This  i< 
the  course  ailnpleil :  the  partial  period*  at  the  lwginnii;i;  ami  end  of  the  entire  seriefl  of 
olix,-r\ations  In-ill^'  oniiti.-.l  The  first  period  actually  employed  was  tli 


26 


to  the  sextant  24o°-3OO°,  in  which  the  first  observation  was  made  on  January  10, 
1847,  and  the  last  on  September  18  of  the  same  year.  The  last  period  corresponded 
to  the  sextant,  i8o°-24o°,  the  last  observation  in  which  was  on  November  13,  1858. 
There  were,  in  all,  ten  periods  corresponding  to  each  sextant,  and  hence  ten  sets  of 
equations,  each  giving  mean  values  of  h,  k,  and  SI  for  periods  extending  through  a  little 
more  than  a  year.  Each  residual  gave  an  equation  of  condition,  for  the  coefficients  of 
which  the  mean  value  corresponding  to  the  entire  sextant  was  taken.  These  values  for 
the  several  sextants  are  as  follow  : 


I 

g 

sin,? 

COSf 

sin'£- 

s\ngcosg 

cosV 

I 

0-60 

+  0.48 

+  0.83 

0.23 

+  0.40 

0.69 

2 

60-  120 

+  0.96 

o.co 

O.QI 

o.oo 

o.oo 

3 

120-  180 

4-  0.48 

-  0.83 

0.23 

-  0.40 

0.69 

4 

180-240 

-  0.48 

—  0.83 

0.23 

+  0.40 

0.69 

5 

240  -  300 

—  0.96 

o.oo 

0.91 

o.oo 

o.oo 

6 

300-360 

-  0.48 

4-  0.83 

0.23 

-  0.40 

0.69 

The  sums  of  the  residual  errors,  corresponding  to  each  period  and  each  sextant 
arranged  in  chronological  order,  together  with  the  number  of  residuals  of  which  each 
sum  is  formed,  are  as  follow: 


Mean 
date. 

»'  =  5 

1  =  6 

>'  =  I 

1=2 

»  =  3 

1=4 

1847.8 

+     6.5 

4-    14-4*1 

4-   l5-4w 

-   ii.7w 

4-     Q.O.J, 

-  16.717 

1848.9 

+     6.7 

+      8.7 

-    33-087 

—     I.Qu 

4-   23.2,8 

4-   3'.5.7 

1850.1 

+     «.  5 

-   34-»i7 

-  40.9*) 

-     9.1.1. 

4-    22.  2M 

4-   33-  Q.". 

1851.2 

4-5    3 

-   59-4i'i 

-   50.7m 

-   23-5*1 

4.8.1 

4-  21-  J.  , 

1852.4 

-   42.8  a 

-   5°.  ON 

-  48.  OIK 

-   2i.5,» 

+    35-0.;,, 

4-   25.4..:, 

I853-5 

-    31-2*1 

-106.9,5, 

—  63-631 

+       1.2,, 

4-     6.0,1 

-  38.  o» 

1854.6 

-   30.317 

-  94  •(>•» 

-  35  -4« 

4-     4-2«8 

4-      I.7n 

-   24.  4m 

1855.8 

-    24.3n 

—  SO.OIB 

7-3*1 

—    6.9,., 

—    22.8|» 

-  4'-°i" 

1856.9 

-    36.2 

-  23.81B 

+    15.  4u 

4-      4-2ar, 

-    48.«« 

-  77-Oi7 

1858.1 

-   54-9-w 

-  48.91-, 

-   56.7^1 

-  47-611. 

-    76.9« 

-  46.2,8 

The  dates  given  in  the  left-hand  column  are  those  corresponding  to  the  mean  of 
each  horizontal  line. 

Patting  .<?,-  for  the  mean  value  of  sin  g  corresponding  to  the  index  i,  as  already 
given;  r^  for  that  of  cos  <r;  and  W;  for  the  corresponding  number  of  observations,  the 
normal  equations  are: 


h 


«,.  r,.)  Jl 


(2  Wi  .v,  e{)  k  =  2  * 
(2  nt  c,.a)   k=2ct 


I  lie  values  of  //  and  k  thus  given  by  the  normal  equations  formed  from  the  system 
of  residuals  shown  in  each  horizontal  line  are  shown  in  the  next  table,  which  also  shows 


87 

tin-  u. i\   111    wliirh    they  are   treated.      Fur  tin-   sake  ol   completeness,  (In-  corie.s|N>nding 
quantities  already    l,,im,|    for  (In-  peiii.d    iSf>j-;j  are   added,  and  included  in  (In-  discus 
i,  \\hicli  n«»w  proceeds  as  follows;  tin-  method  adopted  l.rin^  on.-  wliicli.  though  li'M 
Tons  than  the  former  inn-,  will  chow  in  a  stronger  liylit  (lie  e\  idence  on  which   the 
new   inequality  depends. 

•lie  Imsis  of  tin:  discussion,  we  take  thr  independent  values  ol  A  and  /r,  derived 
from  each  series  «)f  observations,  which  values  an-  uiv<-n  in  tin-  second  and  third  columns 
nl'  the  tahlr.  A  preliminary  coni|>nriMMi  <>r  tin-  first  (WMI  of  film* (1847-58)  with 
the  values  uf  h  anil  k  derived  from  the  formula-  aln-ady  -.'ivi-n  imlical.  >  a  diminution  of 
the  constant  terms  of  those  quantities,  so  that,  instead  of  -f  o".04  and  +  o".28,  thej 
liecome,  iw  a  first  approximation, 

/».  =  +  o".5o 
^  =  +  o".io 

TheM*  constants  are  now  «ul»tracted  from  tin-  \alue>  of  /;  and  /..  I.MMIIL:  a  ^. m  x  ,,i 
residuals  i{i veil  in  the  fourth  and  fifth  columns,  which,  if  (In-  |»n..,lir  term  under  in- 
vestigation has  no  existence,  should  )»•  n-_'.n.|.  .1  .1-  dm:  to  errors  of  oliservatimi,  and,  in 
the  contrary  case,  should  l»e  rapreMntaUfl  liy  the  formula' 

h'  =  —  a  sin  (/i  -f  nt)  -f  accidental  errors 
k  —      a  cos  (p  -f  *0  +  accidental  errors 

To  show  clearly  how  far  they  are  thus  represented,  we  determine  a  coefficient,  a, 
and  an  angle,  .V,  by  the  equations 

a  sin  .V  =  -  //' 
aco«Ar=      k' 

The  next  two  columns  give  the  se\end  \alue>  of  or  and  A*  thus  obtained.  The 
nearly  regular  progression  of  the  niiijl«'  -V  is  too  striking  to  lie  overlooked.  To  see  how 
nearly  this  angle  can  be  represented  as  one  increasing  uniformly  with  the  time,  \\e  so|*e 
the  necessary  eqimlions  of  condition  by  least  sipmres.  It  is  obvious  that  the  urealer  tin- 
value  of  a  the  more  certain  will  be  the  value  of  .V:  we  therefore  uive  weights  propor- 
tional to  or.  Moreover,  weights  nearly  twice  as  great  in  proportion  an-  ^'iven  to  the 
second  series  (1862-74)  as  containing  the  results  from  two  observatories,  and  beinj; 
more  carefully  corrected.  The  values  of  //  and  //  thus  obtained  by  the  method  of  1< 
>'l'iares  are : 

//=|64°.6±4   A 

H=      20    .8  ±0   .47 

The  probable  error  of  a  value  of  .V  of  weight   unity  comes  out 

±33° 

The  residuals  still  outstanding  are  shown  in  the  column  _M  This  value  of  H  \» 
2°  less  than  that  found  from  the  second  series  of  observations  alone,  and  an  examination 
of  the  residuals  shows  that  there  is  a  real  discordance  between  the  \alues  of  (he  angular 
motion  of  .V  sjivcn  by  the  two  series.  It  is  quite  likely  that  the  relative  weights  assigned 


28 


to  the  older  series   of  observations  are  twice  as   great  as  they  should  l>r,  and  that  t In- 
most probable  value  of  the  angle  Arlies  nearly  halt-way  between  the  two  values 


and 


220.8<>-   1868.5) 

20°.8(/-  1868.5) 


found  from  the  last  series  alone,  and  from  the  two  combined.     1  judge  that  the  most 
probable  value  is          . 

.2  +  2i°.6  (t  —  1868.5), 


and  that  the  probable  error  of  the  annual  motion  is  more  than  half  a  degree,  but  less 
than  a  degree.     The  column  4'  N  shows  the  residuals  given  by  this  value  of  N. 


Mean 
date. 

A 

, 

•     *• 

*' 

a 

N 

Wt. 

/'  +  »t 

A  AT 

A'  A" 

1847.8 

—  0.08 

II 

+   0.55 

It 

-  0.58 

N 

4  0.45 

n 
0.74 

.      0 

52 

I 

94 

4    42 

4   24 

1848.9 

-  0.55 

-    1.33 

-  1.05 

-   1.48 

1.82 

'45 

3 

118 

-   27 

-   44 

1850.1 

—   O.2O 

-     I.QI 

—  0.70 

—    2.01 

2.13 

161 

3 

Mi 

—    20 

-   36 

1851.2 

-    0.32 

-     1.92 

—  0.82 

—    2.  02 

2.18 

158 

3 

I6S 

4-     7 

-     8 

1852.4 

+    0.26 

-   2-45 

-  0.24 

-  2.55 

2.56 

175 

4 

189 

+   14 

o 

1853.5 

+     I.  10 

-   1.88 

4-  O.6o 

—  I.  .98 

2.07 

197 

3 

212 

+   15 

4-      2 

1854.6 

4-   1.45 

-   1.40 

4  0.95 

-  1.50 

1.77 

212 

3 

236 

4   24 

+    12 

1855.8 

+  0.77 

+  0.31 

+  0.27 

+    O.2I 

0-34 

308 

i 

26O 

-   48 

-    60 

1856.9 

+    1.  7<> 

+   1.82 

4-1.26 

4    1.72 

2.13 

328 

3 

284 

-  44 

-   55 

1858.1 

-  0.17 

+  0.66 

—  0.67 

+   0.56 

0.88 

5° 

I 

3°7 

-103 

—  112 

1862.5 

+  0.04 

+   1.23 

-  0.46 

4-    1.13 

1.22 

22 

3 

40 

4    18 

4    12 

1863.5 

-  0.64 

+   1.78 

-   1.14 

+  1.68 

2.03 

34 

5 

f)I 

+   27 

4    21 

1864.5 

-   1.07 

4-   1.09 

-   '-57 

4-   0.99 

1.85 

58 

5 

Si 

4-    23 

+    19 

1865.5 

-   1.03 

-  0.15 

-   1-53 

—   0.25 

'•55 

99 

4 

1  02 

+     3 

—     I 

1866.5 

-  0.47 

•+-    O.IO 

-  0.97 

o.oo 

0.97 

90 

2 

123 

4   33 

4-   30 

1867.5 

-  0-93 

—  0.36 

-    1-43 

—  0.46 

1.50 

1  08 

4 

144 

4   36 

4    34 

1868.5 

+  0.34 

-  1.46 

—  0.16 

-  1.56 

1.57 

'74 

4 

165 

-     9 

-    II 

1869.5 

+    1.67 

-  1.56 

4-    I.I7 

-   1.66 

2.03 

215 

5 

185 

-   30 

-   30 

1870.5 

4   1.48 

-  1.14 

4   0.98 

-   1.24 

1.58 

218 

5 

206 

-    12 

—     12 

1871.5 

4-    1.65 

—  0.36 

4-   1.  15 

-  0.46 

1.24 

248 

3 

227 

—    21 

—    20 

1872.5 

+   2.15 

-    0.12 

4-    1.65 

—   O.22 

1.66 

262 

4 

248 

-     '4 

-    12 

>873.5 

+    1.91 

+  0.16 

4-    1.41 

+    O.O6 

1.41 

272 

4 

269 

-     3 

I 

1874.5 

+    1.92 

+  0.60 

+    I.4« 

4    0.50 

1.50 

289 

4 

289 

o 

4-     4 

The  old  and   new  series  of  observations  agree  well  in   giving  for  the   value  of  tin; 

coefficient  of  this  term, 

The,   old   series,  at  —  i". 66 

The  new  series,  a—  i"-55 

The  effect  of  the  accidental  errors  will   be,  on  the  whole,  to  increase  the  value  of 
the  Coefficient.      I  consider  then-litre  that  the  value 

«=i".5o 


89 

m.i\    In-  .lil«p|>tet|   ;i.s  (In-   nii»l    |>i..l..ilili-   \\hirli   .  MII    lie  deliteil    Itolil  all    ill.'  obxen  .(I  Mill- 

It    u.     Militr.irl.  from    each  valiir    of  li    ami    k    in    (lie  precedm;.'    table.   tin-  p.  ii...|i. 

portiuiu 

/*'  =  -  i".50  hiti  [i6j°.2  +  2i°.6  (I  —  1868.5)] 
^=      i".so«»K[i630.2-f  2i°.6(/-  1868.5)] 

ami  take  tin-   mean  value  of  the  ootttanding  remainder  fur  each  series  of  observations 
we  tiinl  it  to  lie  as  follows: 

(  )ld  *cries,  A.  =  +  o".33  ;    *.  =  -  o".  i  7 
New  series.  A,  =  -f-  o".65  ;    A,  =  -j-  o"-36 

The  differences,  o".oi  and  o".o8,  lictween  these  last  valued  ami  those  found  on  page 
23  arise  from  the  different  value,  of  the  |>eriodic  term.  I  consider  that  the  result*  of  the 
second  series  are  entitled  to  three  times  the  weight  of  those  of  the  first,  and  shall  there- 
fore put  for  the  definitive  values  of  It  and  /., 

A  =  +  o".57  +  A' 
*-  +  o".23-M' 
The  correnpoodilig  correetions  to  the  eccentricity  and  longitude  of  |NTigee  are: 


6r  =  +  2.2 

The  corrections  to  the  IIHMHI'-  longitude  arc: 
«5/-c=  —  h  sin  g  —  k  cos  g 

—  —  o".57  sin  g  —  o".23  co«^  +  i".5o  gin  (g  +  JV—  90°). 

The  last  term  U  the  hitherto-unsuspected  inequality  indicated  by  observations,  but  not 
yet  known  U*  be  given  by  theory.  It  may  be  either  an  inequality  of  the  eccentricity  and 
perigee  having  a  period  of  about  i6j)  years,  or  one  of  the  moon's  mean  longitude  having 
a  period  of 


Substituting  first  for  A',  and  then  for  g,  their  values  in  terms  of  the  time,  the  expres- 
sion for  the  inequality  of  longitude  bccomen 

i".508in  [g+  73°-2  +  ai°.6(«—  1868.5)]  =  i".5own  (s6°.8  -f  13°.!  2413  r), 


r  being  the  time  in  days  counted  from  Greenwich  mean  noon  of  1850,  Jan.  o. 

It  would  perhaps  be.  premature  to  introduce  so  purely  empirical  a  term  as  this 
into  lunar  tables  for  permanent  use;  but  where,  as  at  present,  it  is  requisite  to  obtain  the 
corrections  to  the  (aides  during  a  limited  period  with  all  p»issild«-  accuracy,  tin-  evidence 
in  favor  of  the  reality  of  the  term  seem>  >tiun^  enough  to  justify  its  introduction.  The 
only  apparent  cause  to  which  the  term  can  be  attributed  is  tin-  attraction  of  MHIH-  one 
of  the  planets. 

In  the  investigation  of  corrections  to  the  longitude,  it  only  remains  to  determine 
the  slowly-varying  corrections  to  the  mean  longitude,  or  to  /<•*:,  -riven  by  the  ..!.- 
timis      To  determine  the  errors  of  short  period,  we  have  applied  several  emu  dion-  t« 
the  residuals,  not  as  real,  but   only  to  render  the  various  observations  comparable      U 


30 


have  now  to  consider  the  pure  results  of  observations  as  Ihey  would  have  been  had  these 
corrections  not  been  applied.  These  for  the  second  series  of  observations  are  found 
by  taking  the  sum  of  (i)  the  mean  of  the  small  corrections,  applied  on  account  of 
observatory  and  limb,  to  compensate  for  the  systematic  differences  between  results  from 
different  limbs  or  different  observatories;  (2)  general  corrections  to  make  the  residuals 
in  the  mean  very  small ;  (3)  remaining  outstanding  correction  found  by  solving  th(. 
equations  of  condition. 

The  corrections  from  both  series  are  as  follow:    the  corrections  since  1862  may- 
be very  closely  represented  by  a  term  increasing  uniformly  with  the  time,  as  is  shown 

by  the  last  two  columns. 

First  series. 


Date. 

«i'z 

Dale. 

*6s 

1847.8 

-  0.15 

I853-5 

-*•    1-77 

1848.9 

-  0.43 

1854.6 

+    1  .  40 

1850.1 

+   0.32 

1855.8 

+    1.24 

1851.2 

+   1.  13 

1856.9 

+   1.50 

1852.4 

+  0.93 

1858.1 

+   2.40 

Second  series. 


Year. 

(') 

(2) 

(3) 

ntz 

a  +  lit 

A 

1862.5 

+  "-45 

+    2.  10 

i    0.04 

+   2.59 

+   1.52 

+    1.07 

1863.5 

+   0.45 

-+-    1.20 

-  0.27 

1-1.38 

+  0.60 

+   0.78 

I  $64  .  5 

o.oo 

O.OO 

-  0.49 

-  0.49 

—  0.32 

-   0.17 

1865.5 

-  0.15 

-  1.15 

—  0.62 

-   1  .92 

-   1.24 

-  0.68 

1866.5 

-  0.15 

—    2.00 

-  0.75 

—   2.90 

-  2.16 

-  0.74 

1867.5 

-  0.15 

-    3.40 

—  0.41 

-   3-96 

-   3.08 

-  0.88 

1868.5 

-  0.15 

-    4-05 

—    O.2O 

-  4.40 

-   4.00 

—  0.40 

1869.5 

+  0.08 

-    4.85 

-    O.2I 

-   4.98 

-  4.92 

—  0.06 

1870.5 

+  0.08 

-    5.50 

—    0.09 

-   5.51 

-   5.84 

+  o.?3 

1871.5 

o.oo 

-  6-35 

-    0.52 

-   6.87 

-  6.76 

-    O.II 

1872.5 

-  0.15 

-  7.25 

—    0.22 

-   7.62 

-   7.68 

+  0.06 

1873-5 

o.oo 

-  8.30 

+    0.10 

—    8.20 

-   8.60 

+  0.40 

1874.5 

o.oo 

-  9-45 

+    0.38 

-  9.07 

-  9-52 

+  0.45 

IN\  I  >IK.\II(iN    cl     I  UK   I'ol.AK    DIM  \M  I     AM>   I.AII  II   HI 

It  i-  .1  singular  circumstance  that  (luring  the  last  six  years,  at  least,  the  ohsn  \.i 
tiiin>  nl  tin-  inn  m's  pular  distance  are  much  less  accurate  than  those  of  its  right  asccn- 
simi.  Whether  this  is  to  he  attributed  to  the  instruments,  or  whether  it  in  n  result  of 
•  irregularities  in  the  outline  of  the  lunar  globe  in  the  polar  regions,  cannot  at  pres- 
ent \H*  decided.  T*t  whatever  cause  we  attribute  the  errors,  their  existence  renders  a 
rigorous  treatment  ol  the  individual  observations  of  little  value.  We  shall  therefore, 
from  the  whole  of  the  errors  in  declination,  geek  to  obtain  the  liest  corrections  to  the 
inclination  ami  node  of  the  moon's  orbit. 

From  the  derivatives  of  the  moon's  declination   relatively  to  its  true  longitude,  the 
inclination,  and  the  node,  which  have  already  been  given,  we  obtain: 

tt        d&  3ti   i    ^  jr/i    i    d&  »• 
do  —    ,.  <w-f-        00  -4- 

dl       r  dO         r  di 

61  l>eiiig   known  from   the  data  already  given,  the  equations  ol  condition  will   be 
thrown  into  the  form 

'^   •  m  i  *W  »  .        »  ,       dA  t  . 
i  AO  4-  --  dt  —  AS  —       Al 
idtt  itt  dl 


From  the  numerical  expressions  already  given,  we  hav 


tj   dt  =  sec  A  [(0.40  -f  0.08  cos  0)  cos  /  +  0.08  sin  0  sin  /]  «*/ 
dl 

If  we  put 

fi\  —tin-  correction  to  the  moon's  mean  longitude, 
K  —  0.40  +  o.oS  cos  0, 
II  •=.  o.oS  sin  0, 

we  have  the  quantities  of  the  first  order,  with  respect  to  the  i-ccentricilies, 
—  [A"  cos  /  -f-  //sin  f]  [l  -j-  2  rcos  (\  —  IT)  ]  wed 

(I  A. 

The  largest  terms  in  sec  6  arc 

1.040-!-  .016  cos  0  —  .040  cos  2  A  —  .016  !••»  '  .'  A  — 
while,  if  we  replace  /  by  the  mean  longitude,  A,  \\e  skill  h:i\. 

/  =  A  -f  2  e  sin  (A  —  w) 

sin  /  =  sin  A  -f-  f  »»'  (2  A  —  w)  —  r  sin  ir 
cos  /  =  co«  A  +  e  co«  (2  A  —  JT)  —  e  con  JT 

If  we  substitute  these  various  quantities  in  the  expression  for        SI,  we   shall  find 


32 

ho  sensible  terms  depending  en  the  sine  or  cosine  of  the  arguiiu'iit  of  latitude,  A  —  0.     If 

i  &        7  / 

we  substitute  for  SI  its  value  in  S\  we  shall  find  the  principal  terms  in  cos  6  to 

at   dk 

be 

K cos  A  -)-  H sin  A  -(-  3  e,  Kcos  (2  A  —  ?r)  -j-  3  c'  77 sin  (2  A  —  TT) 

In  consequence  of  the  great  number  of  revolutions  of  the  moon  through  which  the 
observations  now  under  discussion  extend,  I  have  considered  that  all  except  the  first  two 
terms  might  be  treated  as  accidental  errors,  which  would  cancel  each  other  during  the 
course  of  the  observations.  Using  for  6\  the  mean  corrections  to  the  moon's  longitude, 
we  have  the  following  values  of  the  correction  to  the  declination  for  those  errors  of 

longitude : 

Year.  Correction. 

//          //  ( 

1862,  -+-  0.9  cos  I  —  0.2  sin  / 

1863,  +  0.6     —o.i 

1864,  -o.i       o.o 

1865,  —  0.6     +  o.  i 

1866,  —  0.8       o.o 

1867,  —o.i     -o.i 

1868,  -  1.4     -  0.2 

1869,  -  1.8     -o.i 

1870,  -  2.2        -  0.4 

1871,  —  2.8     -  0.6 

1872,  —3.3     -0.6 

1873,  -3.8     -0.5 

1874,  -4.2     -0.4 

The  mean  correction  to  the  moon's  tabular  north-polar  distance  for  each  year,  from 
observations  of  each  limb  at  each  observatory,  was  taken  with  a  view  of  detecting  any 
constant  error  of  sufficient  magnitude  to  affect  the  final  results  for  errors  of  the  node 
and  inclination.  These  means  should  have  been  taken  after  the  application  of  the  cor- 
rections just  found:  actually,  however,  they  are  the  mean  corrections  given  by  the 
observations,  tiller  applying  the  following  constant  corrections  to  reduce  the  declinations 
to  the  same  fundamental  standard  : 

To  Greenwich  observations  of  N.  P.  D.  To  Washington  observations  of  N.  P.  D. 

1862-67,     -0.4  1862-65,    -f  0.5 

1868-74,      +0.2  1866-67,        -I.I 

1868,  -  I   2 

1869,  —  0.6 

I  <S 70-7 2,     —0.4 

1873-74,        o.o 

These  corrections  are  approximately  those  necessary  to  reduce  the  slur-observa- 
tions of  the  several  years  to  Auwers's  standard  of  declination.  The  change  in  the  Green- 
wich correction  between  1867  and  1868  probably  arises  from  the  introduction  of  a  new 


enn-tant  .if  refraction  in  1868,  while   tin-  change  in  tl,e  Washington  correction  in    1866 
corre*|.,.n.!*  In  (lie  introduction  of  the  lame  transit  circle  in  plan- ,,(  thr  ul.l  niunil  cirri.-. 


Correction  to  N. 

'. 

Greenwich. 

1  • 

N.  L. 

S.  U 

N.  U 

S.L. 

X 

M 

»» 

M 

1861 

—   O.I 

—  o. 

-  o.j 

—  O. 

1663 

+  o.« 

—   0. 

-  o.$ 

—  1. 

1864 

+  0.4 

-   0. 

+  0.8 

—   0. 

1865 

-«•  o.j 

—  o. 

+  1.1 

—  o. 

1866 

-  0.7 

—   0. 

+   «-4 

—  0. 

1867 

-  0.4 

—   0. 

+   O.I 

-  1. 

1868 

-  0.7 

—  1. 

+  o.i 

-1-  0. 

1869 

—   O.I 

—    . 

-  0.8 

—  I. 

1870 

-  0.6 

—    . 

—  O.I 

—  1. 

1871 

—  o.i 

—    . 

+  fl.i 

—  1. 

•872 

0.0 

. 

-  0.7 

—   0. 

•«73 

-  0.9 

+    . 

+    1.0 

—   0. 

t»74 

•      • 

•      • 

-  1-7 

-  <M 

The  large  residuals  of  the  Washington  observations  of  the  south  limb  led  to  the 
application  of  the  further  systematic  correction  of  -f  i".o  to  all  those  observations  before 
combining  them  all.  The  corrections  arising  from  the  error  of  mean  longitude  were 
then  applied,  and  the  outstanding  residuals  were  considered  to  arise  from  accidental 
errors  and  from  errors  of  the  inclination  and  mule.  The  equations  of  condition  thus 

>mc 

0.92  sec  6  [sin  (/  —  6)  8i  —  cos  (/  —  0  )  i  69]  —  66 
or 

sin  (/—  0)  6i  —  cos  (/  —  6)  i  60  —  1.09  cos  6  X  ** 

( >\\  ing  i<i  the  smullness  of  the  final  residuals,  <$<?,  the  factor  1.09  cos  6  may  be  consid- 
ered as  a  constant,  and,  in  the  actual  solution,  has  been  put  equal  to  unity.  Its  mean 
value  is  more  exactly  1.04,  and  its  effect  may  be  obtained  by  dividing  the  final  results 
I iy  this  factor. 

The  final  values  of  the  residuals  were  then  arranged  according  to  the  values  of 
A  —  0,  or  the  moon's  mean  argument  of  latitude,  as  the  residuals  in  right  ascension  were 
arranged  according  to  the  mean  anomaly.  The  sum  of  the  residuals  corre8|K>nding  to 
each  interval  of  20°  in  the  argument,  with  the  corresponding  number  of  observations 
fur  each  year,  is  shown  in  the  following  table : 


34 


Sums  of  errors  of  the  moon's  corrected  declination,  f/iren  Inj  observations  at  GreeinrirJ/ 

and  Washington. 


1862. 

1863. 

1864. 

1865. 

1866. 

1867. 

1868. 

2(5,5 

N. 

SiM 

N. 

Zdd 

N. 

2(5(5 

N. 

£<5(5 

N. 

£<M 

N. 

2<W 

N. 

0         0 

,, 

,, 

II 

H 

a 

„ 

„ 

o  to  20 

-  3-8 

8 

+  1.3 

3 

+  4-° 

8 

+  5-4 

9 

+26.7 

ii 

-  2-5 

8 

+  0.4 

9 

20  to  40 

+  9-6 

9 

+  5-8 

7 

-  0.4 

9 

+  6.0 

7 

+  2.6 

12 

-  2-3 

9 

—  6.1 

17 

40  to  60 

-  1-4 

9 

+  6.9 

10 

+  6.5 

6 

+  9-7 

7 

+  4-0 

9 

-  4.9 

10 

-  7-9 

'5 

60  to  80 

o.o 

7 

+  16.4 

10 

+  6.6 

8 

+  8.7 

12 

—  I.I 

16 

+  14.5 

ii 

-  0.3 

5 

80  to  100 

+  8.6 

II 

+  0.4 

12 

+  11.  1 

6 

+  7-9 

II 

+  5-5 

7 

+  0.5 

10 

-12.8 

12 

100  to  1  20 

+  3-2 

7 

+  8.5 

15 

+  3.2 

5 

+  7-4 

7 

—  I.O 

8 

-  6.1 

6 

+  0.2 

6 

120  tO  140 

-  9-2 

12 

+  3-1 

8 

—  6.1 

8 

+  0.8 

ii 

+11.9 

14 

-12.4 

8 

-  6.8 

ii 

140  to  160 

-  0-3 

4 

-  4.6 

9 

—  2.2 

5 

-  9-7 

15 

—  1.2 

10 

-  7-7 

12 

+  6.2 

14 

160  tOiiSo 

+  0.5 

9 

—  10.4 

6 

—  IO.4 

12 

+  0.5 

9 

+  2.2 

10 

-  8.9 

9 

—  II.  2 

9 

1  80  to  20O 

-  "8.6 

6 

-  5-7 

ii 

-  0.6 

7 

-  5-3 

12 

-  7-7 

6 

-15.2 

'4 

+  3-1 

10 

2OO  to  22O 

—22.3 

8 

-II.  6 

10 

+  4-7 

12 

-  5-4 

9 

-  3-3 

TO 

-  6.8 

14 

-ii.  8 

ii 

22O  to  24O 

-14.4 

12 

—  IO.2 

9 

-  8.8 

10 

—  I.O 

7 

—  2.0 

'3 

-  5-9 

12 

—  IO.O 

13 

240  to  260 

-12.4 

7 

—  12.3 

9 

-  4-1 

8 

+  4.6 

ii 

+  1.2 

9 

-  0.6 

9 

-  9.2 

15 

260  to  280 

-  2.3 

4 

-  3-2 

4 

-  8.5 

8 

+  1.5 

9 

-  5-3 

9 

•+•  I.O 

8 

+  I.  g 

9 

280  to  300 

—  2.2 

7 

-  4-3 

8 

-  8.4 

II 

-  4.0 

4 

-  3-5 

9 

-ii.  4 

'3 

o.o 

'3 

300  to  340 

-  7-1 

10 

-  6.2 

10 

•4-  9.6 

8 

+  3-1 

5 

—  O.I 

13 

-  8.4 

10 

+  0.4 

8 

320  to  340 

+  2.0 

7 

+  3-4 

8 

+  6.0 

12 

+  8.6 

6 

+  8.4 

ii 

+  2.1 

9 

-  6.7 

'4 

340  to  360 

+  7-3 

5 

-  6.5 

5 

+  4-9 

13 

+  11.  6 

8 

+  7-0 

14 

+  2.9 

3 

-  3-1 

12 

—84.0 

142 

-75-0 

'54 

—49-5 

I56 

-25.4 

159 

—  25.2 

191 

-93-1 

'75 

-85.9 

203 

+31.2 

+45-8 

+  56.6 

+75-8 

+  69.5 

+  21.  0 

+  12.2 

-52.8 

—29.2 

+  7-i 

+  50.4 

+44-3 

-72.1 

-73-7 

Sums  qf  errors  qf  tke  WMMN'M  corra-/**/  »/» « /imi/ion,  «(<  — ( '••niiun.  <1 


1869. 

1870 

. 

I87l 

• 

1872 

•  •73 

• 

•  •74 

• 

IM 

N. 

1*1 

N. 

sw 

N. 

: 

N. 

ZM 

N. 

XM 

N. 

•     • 

" 

»• 

H 

a 

** 

M 

o  lo    so 

7-1 

+             JL 

• 

4-        7-7 

40  lo   to 

4-   II.  S 
4-     6.4 

7 

•  O 

4-     .6 

10 

-  0.8 

u 

-    1.2 

10 

-  S.S 

i 

/•S 
-     17.0 

u 

to  lo    80 

-     S-o 

9 

+     •$ 

7 

4-I3-* 

ii 

-  s.i 

7 

-IJ.7 

7 

-  «S-4 

14 

100  10  ISO 

—    s.o 
-  «3-7 

u 
7 

4-      .S 
-      .S 

9 

IS 

4-13.1 
-  6.3 

9 

—  S.o 

8 

-    1.0 

9 

-  11.4 

11 

iso  to  140 

-  II-  4 

9 

+  4-5 

7 

-  1-9 

S 

4-  O.S 

14 

-  1.1 

4 

4-      S.I 

6 

I  to  to  160 

-  15-4 

IS 

-11-7 

u 

-  4-6 

-8.9 

9 

4-   7-4 

5 

-    11.6 

u 

ito  to  180 

-     »•$ 

II 

-  S-7 

•3 

-  S-1 

9 

-  4.6 

12 

-  39 

7 

-     S-9 

180  |O  MO 

200  10  ISO 

—     5-4 
-     5-4 

9 

4 

—  0.$ 
-IO.S 

IS 

+•  5-4 
-  6.S 

10 

9 

+  $•« 

7 

-   19-3 

12 

33010140 

-    6.6 

6 

—  I.I 

9 

4-  9-1 

14 

-  4-S 

10 

-  «-7 

S 

-    !$•« 

6 

14010  ito 

-  18.4 

IS 

—II.  I 

8 

+  S-6 

8 

-H3-7 

•3 

4-14-8 

11 

4-6 

6 

•to  10180 

-     7-7 

7 

-15.4 

IS 

-  6.S 

7 

4-  S.S 

u 

4-S0.7 

9 

-     l.S 

• 

18010  300 

-   ll-4 

13 

—  10.  1 

5 

+  3-6 

8 

+  S-« 

9 

+  3-3 

10 

-   s-s 

0 

30010320 
3soto3«o 

34010360 

+     S-3 
+     5-7 
o.o 

7 
S 
10 

-10.3 

7 
IS 

8 

9 

-  1-3 

—  IS.S 

II 
IS 

+  4-3 
4-14-0 

8 

7 
II 

4-     0.3 
4-     4-0 
-     3-9 

10 
10 

9 

-104-9 

•55 

—93.6 

166 

-47-1 

153 

-67.4 

167 

-4S.6 

140 

-IS4-3 

169 

+  35-7 

4-33-I 

+S3-8 

4-SO.S 

+90.7 

4-   14  1 

-  to.i 

-$9-5 

•»-  6.7 

-47-  » 

_+*•_' 

-140.1 

The  general'iiTegularity  of  the  residuals  in  declination  is  such  that  no  great  a«lvan 
tage  would  result  in  a  separate  solution  of  the  equations  for  the  separate  years.     Tin- 
sum  of  the  residuals  for  each  20°  of  the  argument  wa*  therefore  taken  during  the  whole 
thirteen  years  of  observation,  with  the  following  result: 


l-» 

ZAJ 

N. 

%-« 

ZA4 

N. 

•               • 

olo   so 

•• 

4-  47-  « 

i<>3 

•          • 
18010  soo 

-  62.3 

110 

solo   40 

4-   10.7 

i"S 

SOOIO  330 

-  9S  3 

138 

4010   to 

4-     6.1 

119 

330  U>  *40 

-  73  3 

126 

to  w   80 

4-  is.  3 

1*4 

140  to  sto 

-  3».» 

137 

8010  too 

4-  34-3 

ill 

36010  380 

-  23.8 

106 

1     looioian 

-  36.1 

ill 

380  to  300 

-  so.i 

1*3 

i    iso  to  140 

-  »7-4 

iso 

300  to  320 

-     5-4 

»S 

14010  ito 

-  65.3 

U4 

jso  to  340 

4-   19-  » 

ISS 

160  to  180 

-  65.4 

IS6 

54010360 

4-  SO.S 

no 

36 

Leaving  in  the  equations  a  constant  term  Sp,  representing  the  mean  constant  error 
still  outstanding  in  the  measures  of  declination,  the  solution  of  the  equations  of  con- 
dition given  by  the  residuals  gives  the  following  results  : 

dp  —  —  o".  1  7 


Correction  to  the  inclination,  —  o".i5 

Correction  to  the  longitude  of  node,  +4"-5 

This  correction  to  the  longitude  of  the  node  from  Hanson's  tables  implies  a  dimi- 
nution of  the  secular  retrograde  motion  of  the  node,  which  is  quite  accordant  with  the 
results  derived  from  ancient  eclipses.  Hansen  remarks  that  an  increase  of  1  2"  per  cen- 
tury in  the  longitude  of  the  moon's  node  will  improve  the  agreement  of  his  tables  with 
ancient  eclipses;*  and,  if  we  suppose  the  tabular  longitude  of  the  node  to  have  been  cor- 
rect in  1825,  this  would  imply  a  correction  of  +  5".  2  to  the  longitude  of  the  node 
in  1868. 

*  Darlegung,  etc.,  Th.  ii,  p.  391. 


87 


\i\lMAk\     i.u.i  |    FACILITATING    nil    COMPUTATION    <>i     nil    COR  EEC 

ini\>  TO  M.\\-I  W8  "TABLES  M    l.\  I  (  \i    .  (.i\  IN  \-.\    un    n<i  CEDING  D13 

Tin-  following  is  a  summary  of  the  corrections  to  (In-  longitude  of  (hi-  moon  from 
ll.niM-ii's  tables  given  by  tin-  preceding  discussion.  'I'll.-  first  >ix  t.-rms  an-  applicable 
to  the  «listurli«-.l  mean  longitude,  or  "Argument  foudumfntal";  tin-  remainder  to  Un- 
true lonjjitu.le;  but  they  may  all  \tc  used  as  corrections  of  the  "Argument  fon<l<imcntal" 
without  serious  error: 

OIM  on  account  of  diminution  of  the  .so//;/  jxinillax.  .    n  6;  =  +  o".g6  sin    l> 

+  o".07.  si  .. 

-o".  1  3  sin 
On  account  of  hypothesis  (ft-  mionally  set  «.«/Wr),  ////// 

the  moon's  center  of  grarity  Joes  not  coinci<t>  iritli  th, 
tcr  ofjigure,  together  irith  th<-  correction  to  the.  evec- 
tio*  resulting  from  the  correction  to  theecc<-n//i<iti/.  .  .  n  6z  =  +  o".O9sin  g" 

-o".338in  2  It 

—  o".2i  sin  (2  D  —  g) 
OH  account  of  term  accidentally  intru>lun<l  into 

the  taltlix  irith  a  wrong  fign  ..........    6v  =  —  o".62  sin  (2  g  —  4  gf  +  2  w  —  4  u') 

OH  account  of  corntfi<,n  to  the  eccentricity  and  jn-rigee 

found  from  observations  during  1847-74  .......    do  —  —  o".tf8\ng  —  o".2^ctwg 

=     o".62  sin  (g  -|-  202  °.o) 
Empirical  term,  neceuary  to  satisfy  observations, 

but  not  verified  by  theory  ................   +  i".5O8in  [g  -f  2i°.6  (  Y—  1865.1)] 

Unexplained  correction  to  the  mean  longitude,  changing  slowly  from  year  to 

year  .......  ....................................  See  Table  I  V. 

The  deduction  of  all  these  terms,  except  the  last,  has  been  fully  given  in  the  pre- 
ceding pages.  This  secular  correction  to  the  mean  longitude  has  been  derived  from  the 
outstanding  errors  of  mean  longitude  given  on  page  30,  in  the  column  n  62,  by  suppos- 
ing this  quantity  to  vary  according  to  some  simple  law,  which  law  changes  when  necessary, 
so  as  to  satisfy  the  observations  within  the  moan  limits  of  their  probable  error.  An 
examination  of  Table  IV  shows,  that,  from  1848.0  to  1855.5,  tn<>  correction  is  sup|»nsed 
to  increase  uniformly  at  the  rate  of  o".2O  per  annum.  It  is  then  supposed  to  remain 
constant  until  nearly  1863.0,  a  i>eriod  during  which  the  observations  are  not  continiio 
there  being  no  comparisons  with  theory  from  1859  to  1861  inclusive.  From  1863.0 
until  the  present  time,  the  observations  are  well  represented  by  the  correction 

-  5"-53  -o".86  (t  —  1870.0)  +  o".02  (t  —  1870.0)* 
The  continuance  of  this  correction  beyond  1875.0  is,  of  course,  purely  conjectural. 

TABLES  FOR  APPLYING  THE   PRECEDING  CORRECTIO 
Tin-  following  tables  are  designed  to  facilitate  the  computation  of  the  correction!! 


38 

just  given.  To  avoid  the  necessity  of  referring  to  Hansen's  tables,  the  values  of  all  the 
necessary  arguments  are  given  for  the  years  1850  to  1889  in  Tables  I  to  III. 

Table  I:  the  epochs  are  January  o,  Greenwich  mean  noon  of  common  years,  and 
January  i  of  leap  years.  All  the  arguments  increase  uniformly  by  a  unit  in  a  day. 

Argument  g  is  the  moon's  mean  anomaly,  converted  into  days  by  dividing  its  ex- 
pression in  degrees  by  13.065.  It  is  equal  to  Hansen's  argument  g  diminished  by  1  5  days. 

Argument  D  shows  the  number  of  days  since  mean  new  moon,  or,  it  is  the  mean 
departure  of  the  moon  from  the  sun  expressed  in  days.  It  is  equal  to  Hansen's  argu- 
ment 33  diminished  by  30  days,  or,  which  amounts  to  the  same  thing,  by  od47. 

Argument  A  gives  the  number  of  days  from  the  time  when  the  angle 

2g  —  4g'  +  2(0  —  400' 

was  last  zero. 

Argument  B  is  that  of  the  empirical  term  indicated  by  observations,  but  not  given 
by  theory. 

Argument  u  is  that  of  latitude,  or  the  number  of  days  since  the  mean  moon  last 
passed  her  ascending  node. 

Tables  II  and  III  do  not  seem  to  need  explanation.  In  using  the  former,  care  must 
be  taken  to  diminish  by  one  day  the  dates  for  January  and  February  of  leap  years. 

Table  IV  gives  the  secular  corrections  to  the  mean  longitude,  or  to  ndz,  obtained 
from  observations  in  the  manner  already  described. 

Table  V,  argument  A,  gives  the  correction  for  the  term  introduced  into  the  tables 
with  a  wrong  sign,  described  on  page  9.  It  is  properly  to  be  applied  to  the  true  longi- 
tude, and  is  therefore  designated  as  Sv. 

Table  VI  gives  the  empirical  term,  which,  so  far  as  is  known,  may  be  applied  to  the 
true  longitude. 

Table  VII  gives  the  sum  of  the  terms  of  mean  longitude 

+  o".g6  sin  D 
—  o".33  sin  2  D 
-o".  1  3  sin  (#  +  #') 
+  o".09  sin  g' 

The  sun's  mean  anomaly,  g',  having  a  period  of  a  year,  the  sum  of  these  terms  can 
be  expressed  as  a  function  of  D  and  the  month,  and  is  given  in  the  table  for  the  middle 
of  each  month,  and  for  each  day  of  D. 

Table  VIII  gives  the  sum  of  the  terms  of  true  longitude  which  depend  wholly  or 
partly  on  the  moon's  mean  anomaly,  namely: 

+  o".62sin(£+202°.o) 
+  o".o;  sin  (D  -  g) 


—  o".2i 


The  sum  of  the  terms  of  n  dz  are  to  be  reduced  to  corrections  of  the  longitude  in 
orbit  by  multiplication  by  the  factor 


i  -f  2  e  cos  g  -f  ?  e*  cos  2  g] 


This  factor,  less  unity,  is  given  in  Table  IX. 


eunxenience,  lii,-  unit  (it  tin-  larlor  is  omitted  Imiu  tin-  tabular  number*,  no  that 

»nly  necessary  to  add   tin-  product  f'X  «  <*-  i»  with  n  6:  and  dr  to  ha\e  tin-  cor- 
,"ii  ni'  tin-  true  longitude  in  orbit 

'I'h. -si-  corrections  beiim  applied  to  tin-  lonailiide  ul'  the  moon's  center  I'miii  Han- 
I  tallies,  that  longitude  may  In:  regarded  as  correct,  excepting  n  small  correction, 
which  ma\  probably  he  regarded  as  constant  during  any  one  |M>riod  not  excocdmi:  si\ 
luuntlis,  ami  which  may  he  attrihutcil  to  tin-  adopted  position  of  tlic  equinox.  It  \\ill 
bo  best  determined  from  orcultations  nt  stars  observed  at  points  whose  longitudes  from 
Greenwich  arc  accurately  known  by  telegraph,  and  will  then  be  applicable  to  the 
determination  of  the  longitude  of  any  station  from  occultationa. 

If  the  corrections  here  deduced  are  applied  to  the  errors  of  the  lunar  ephenieris 
deri\ed  from  meridian  observations,  it  must  be  remembered  that  these  observations  are 
made  on  tin-  moon's  limb,  while  the  corrections  are  applicable  to  the  center.  Hence, 
the  value  of  the  moon's  semi-diameter  must,  if  great  refinement  is  aimed  at,  be  varied 
with  the  observer,  the  instrument,  and  the  brightness  of  the  sky.  For  large  instru- 
ments, Hanson's  semi-diameter  is  about  i  too  great,  even  at  night 

The  sum  of  all  the  terms  of  n  <Jr.  <Jp,  and  FX"  <f-  fr°tn  *»c  tables  will  lie  the 
correction  of  the  longitude  in  orbit.  This  will  not  be  rigorously  the  same  as  the  correc- 
tion to  the  ecliptic  longitude. 

Table  X  gives  the  small  factor  (f./)  by  which  the  orbit  longitude  mu-i  IK-  increased 
or  diminished  to  obtain  the  ecliptic  longitude.  This  factor  may  generally  be  disregarded. 

Table  X  also  gives  the  data  for  the  correction  of  the  moon'-  latitude,  namely,  ( i ) 
a  factor  (F.  ft)  by  which  the  correction  of  the  moon'.-  argument  of  latitude  mu*t  he 
multiplied;  and  (2)  the  term 

6ftt  =  —  o".i5  sin  u 

arising  from  the  correction  to  the  tabular  inclination  of  the  moon's  orbit  The  correc- 
tion of  the  moon's  argument  of  latitude  heing  that  of  her  longitude,  minux  the  correction 
of  her  node,  the  whole  correction  to  the  latitude  will  be 

<J/?  =  <5/?,  +  (F./?)(«Ji;-4".5) 

Table  XI  gives  the  factors  for  convortinir  corrections  of  longitude  and  latitude  into 
corrections  of  right  ascension  and  declination.  The  formula'  are 

S.jfi    —dv  +  (v.d)6v  +(ft.a)  6ft 
6 .  Dec.  =  <?/*+  (r .  <J)  <5r  +  (ft.  <J)  6ft 

The  side  argument  is  the  moon's  longitude,  and  in  the  (  oeflicients  («.<*)  and  perhaps 
(ft.  a)  regard  must  be  had  to  the  moon's  latitude  also.  Three  columns  are  therefore 
for  latitude,  —5°,  o°,  and  -f  5°  respectively. 


40 


As  an  example  of  the  use  of  the  tables  of  corrections,  we  will  commence  the 
determination  of  the  corrections  for  September,  1874.  We  find  (he  values  of  the  argu- 
ments for  September  i,  from  Tables  I  to  III,  as  follows: 


S 

D 

A 

B 

u 

1874  .     .     . 
Sept.  I     .     . 
Periods  .      . 

Arg.  Sept.  I  . 
Arg.  Oct.  i  . 

5-4 
23.6 
—  27.6 

12.  1 

7.8 

8.0 
1.9 

20.  o 
24.6 
-27.4 

1.9 
26.2 
—27.2 

1.4 

19.9 

9.9 

17.2 

0.9 

3-8 

20.3 

39-9  \ 
or     7.8  ) 

47-2) 
or  19.8  f 

30.9) 
or    3.7^ 

.0=19.9 
g=    1-4 

D-g* 

:.8.5 

The  tabular  numbers  are  then  found  as  follows,  with  an  argument  increasing  by 
unity  each  day.     From  Table  VIII,  we  take  a  mean  from  columns  18  and  19. 


September     . 

i. 

2. 

3- 

4- 

5- 

6. 

7- 

8. 

,, 

,, 

H 

II 

,, 

„ 

,, 

„ 

Table  IV  («  iz)   . 

-  9.11 

-    9.II 

-  9.11 

-  9.11 

-  9.12 

-  9.12 

—    9.  12 

-  9.12 

V  (<fo).      . 

-t-  0.40 

+  0.55 

+  0.62 

+  o  59 

+  0.48 

4-     0.28 

4-   0.05 

—  0.18 

VI  (<!z<)     . 

-  1.07 

—   1.29 

-   1.42 

-   1.50 

-  1.48 

-   1.40 

-    1.23 

—     1.  01 

VII(»As). 

—  1.29 

-  1-25 

—    I.  12 

-  0.95 

—  0.76 

—  0.56 

-  0.37 

—    O.22 

VIII  (Av)  . 

—  0.65 

—  0.72 

-  0.77 

—  0.78 

-  0.75 

—   0.69 

—  O.6o 

—    0.48 

-11.72 

—  U.82 

—  i  i  .  80 

-11.75 

—  u  .  63 

-11.49 

—  11.27 

—  II.  01 

ntsXF,  Table  IX 

—    I.  12 

-  0.93 

-  0.71 

-  0.47 

—  0.28 

—  0.06 

+  0.18 

4-  0.37 

6v  

—  12.84 

—  12.75 

—  12.51 

—  12.22 

—  11.91 

—  11.55 

—  11.09 

—  IO.&4 

I'P  —  4".5  .      .      . 

-17-3 

—17-  a 

-17.0 

-16.7 

—  16.4 

—  16.0 

-15.6 

-I5.I 

Table  X  (F  .  /?)  . 

4-    0.088 

+  0.082 

+  0.070 

+    0.056 

+  0.038 

+  0.019 

—    0.002 

—   O.022 

X(d/?,).     . 

—    0.03 

—  0.07 

—    O.IO 

—   O.I2 

-  0.14 

-  0.15 

-    0.15 

-   0.14 

(*--4".5)(F.ft- 

-    1-52 

-  1.38 

-  1.19 

-   O.Q3 

—  0.63 

—  0.30 

4-  0.03 

+  0.33 

AS  . 

—    1.55 

1.45 

—  1.29 

—    I.O5 

—  0.77 

—  0.45 

—    O.  12 

4-  0.19 

0 

« 

• 

• 

o 

e 

0 

e 

D's  longitude     . 

46.5 

60.7 

74.6 

88.1 

101.5 

II4.5 

127.4 

140.0 

II 

n 

II 

II 

II 

it 

,/ 

" 

(l+(v.a)){v.     . 

-13.08 

-13.49 

-13.84 

-13.81 

-13.39 

—  12.  6l 

—  1  1  .  62 

—  10.67 

UM'V*     .      .      . 

+  0.47 

4-  0.32 

4-  0.16 

4-   O.O2 

—  0.07 

—    O.Og 

—  0.03 

4-  0.06 

d/R                       j 

—  12.6 

-13.* 

-13-7 

-I3-8 

-13.5 

-12.7 

-ii.  6 

—  10.6 

-o".84 

-0-.88 

—  o".  91 

—  O*.92 

—  o"-9O 

-o«.85 

-o".78 

-0-.7I 

II 

II 

a 

H 

u 

a 

" 

•• 

(v.fyiv     .     .     . 

-  3-70 

—     2.64 

-    I.4I 

—    O.l8 

+    1.02 

+  2.05 

4-  2.84 

4-  3-35 

(l+/S.rf)<!/5    .     . 

-  1.50 

-      1.42 

-    1.28 

-     1.05 

-  0.77 

-  0.44 

—    O.I2 

4-  0.18 

c)Dec  

-  5-2 

-  4-1 

-  2-7 

—    1.2 

+  o.a 

4-  1.6 

+    2-7 

+  3-5 

41 


This  <•  im|iiil.itiini    ln>    I.--  -n    fiintiiitii-.l    In  1X75,  .F.imi.  try    ;i.  mnl    tlir    r«-j.iili» 

-ll.'U  II   ill   till-   liil|ii\Mliy  t;il. 


Corrections  to  th«    /•.'/<A<-mero  dtriveil  fr»  m  //<i«vo/\  '/',,/,/,>  <>i'  th<    M'.<»i.   t 

of  each  tltiy,from  1874,  Srpteiuber  i,  to  1875,  Jnnn,n->i  \\. 


Dale. 
Gr.  mean 

DOOM. 

Correction  to  tabular— 

in    mran 
noon. 

C< 

Lai. 

to  Uhula 

R.A. 

r— 

Long. 

Ut. 
-  1.6 

R.A. 
-ia.6 

Dec. 

-  s.a, 

Dec. 
+  3-6 

1874. 
Sept.      I      -n  8 

•  874- 
Oct.     1  1 

H 

-  7-5 

** 

+  I.I 

-  6.7 

l 

ta.8          1.3 

13.  t 

4.«    i 

7-« 

I.o 

6.8 

3  a 

3 

. 

ia.$          1.3 

l*.t               1.0 

13-7 
13.8 

a.  7                «3 

-  i.a  i1             14 

6.9 
6.6 

I.O 

0.8 

6.8 
7.0 

a.  7 
a.o 

7 

11.9 

-11.6 
II.  1 

0.8 
-  o.$ 

—  O.I 

•3-5 

-It.  7 
tl.6 

+  o.a  |            15 

+    1.6  I                 16 

•  7  I             «7 

6.4 

0.7 

7-0 

•  J 

6.1   i        0.3 

—  o.a 

a 

IO.6       +0.3 

10.6 

35  |              «8 

6.3 

68 

I.O 

9         10.  i          0.5 

9.6 

4-1    ,                «9 

6.4 

—   O.I 

6.6 

1.8 

to          9.6          0.7 

8.6 

43                   «o 

6.8 

0.4 

6.6 

a.  5 

1  1 

—  9.0    +  0.9 

-  7-9 

+     -4  !              »« 

-  7-5 

-  06 

-69 

-  3-3 

n 

8.3          i.o 

7-3 

.a  l]            aa 

8.3 

0.9 

7-3 

41 

13 

7.6  j        I.I 

6.7 

.8                a3 

9-3 

1.1 

B.I 

4-7 

M 

6.8           l.o 

6.a 

•3  j|               »4 

«o-4 

••3 

9-a 

5-a 

i$ 

16 
17 

6.  a  '        i.o 

5-9 

•8 

as 

. 
»7 

11.4 

-ia.4 
13.  a 

••5 

-  «-S 
1*4 

10.6 

-la.a 
•3-9 

5-a 

-  4.8 
3.8 

5.  a          0.7 

55 

1.6 

:                 5.0             0.6 

5-5 

I.O 

aS 

•3.6 

i.a 

ll.l 

a-3 

'9 

$.1          0.4 

5-8 

•*-  0-3 

- 

13.8 

I.O 

•5-7 

-  06 

•0 

$.4     +  o.s 

6.1 

-  0.4 

13.6 

0.6 

•5-3      •    '   i 

H 

—  6.1          o.o 

-  6.6 

—  i.a 

3« 

-13.  a     -  0.3 

-14.  a 

+  a.6 

7.0     —  o.a 

7-* 

a.  a 

Nov.      i 

13.4        +    O.I 

13.6 

36 

13          8.1 

o.$ 

7-8 

3-* 

S 

11.4           0.4 

n.  t 

43 

•4 

9.4          0.8 

8.5 

4-3 

3 

10.5          0.7 

,6 

4-5 

as 

10.6           1.1 

9-3 

5-* 

4 

9.5          0.8 

8.4 

4-5 

a6 

-11.8     -  1.3 

-10.3 

-  5-8 

5 

-  8.J 

+    I.O 

-  7-4 

+  4-3 

t7 

ia.7 

1.1 

u.  S 

6.0 

7.7          i.o 

6.7 

39 

38 

'3-4 

1.6 

IS.  8 

5-7 

7 

7.1          i.o 

6.3 

3-5 

- 

«3-7 

«•$ 

13-9 

4-8 

8 

6.6 

l.o 

6.3 

3  • 

30 

13.8 

1.4 

•4-9 

3-4 

6.4 

I.O 

6.3 

a.6 

Oct.       i 

-»3-S 

—  t.  a 

-i$.a 

-   t.7 

to 

-  6.3 

+  0.9 

-  6.$ 

+  a.i 

* 

13.0 

0.9 

14.7 

—  o.t 

n 

6-3 

0.7 

6.9 

•  •5 

3 

ta.a 

o-$ 

13.6 

+    1-4 

13 

6-5 

0.6 

73 

*  0.8 

4 

it.  5 

—  O.I 

ta.a 

a.  6 

«J 

6.8 

0.4 

7-7 

—   O.I 

S 

10.6 

+•  O.I 

to.8 

3-3 

»4 

7-« 

+    O.I 

7.8 

I.O 

6 

-  9-9 

+  0.4 

-  9-4 

+  3-9 

•5 

-  7-4 

—   O.I 

-  7-8 

-  t.8 

7 

9-» 

0.6 

1.4 

4-1 

16 

7-7 

0.4 

7-7 

a-7 

1 

••7 

O.I 

7-7 

4-  » 

•  7 

8.1 

0.6 

7-6 

3-4 

9 

8.  a 

I.O 

7-a 

l-a 

18 

8.4 

0.8 

7-6 

4.0 

to 

7-8 

I.O 

6.8 

4.0 

•9 

8-9 

I.O 

7-8 

4-5 

I  • 


42 


Corrections  to  the  EpJiemeris  derived  from  Hansen's  Talks  of  the  Moon,  etc.— Continued. 


Date. 

Correction  to  tabular  — 

Date. 

Correction  to  tabular  — 

Gr.  mean 
noon. 

Long. 

Lat. 

R.  A. 

1 

Dec. 

Gr.  mean 
noon. 

Long. 

Lat. 

R.  A. 

Dec. 

1874. 
Nov.    20 

-   9-4 

-     I  .2 

-    8.2 

-   4-8 

1874. 
Dec.    27 

[0.8 

+  0.6 

—  IO.2 

+   4-3 

21 

IO.O 

i-3 

9.0 

4.8 

28 

10.4          0.8 

9-5 

4-7 

22 

10.6 

1.3 

10.  I 

4-5 

29 

10.  I 

I.O 

8.9 

4-9 

23 

II.  2 

1.3 

II.  4 

3-8 

3° 

9.6 

1  .2 

8.4 

4-9 

24 

II.  7 

1.2 

12.7 

2.6 

3' 

9.2 

1.2 

8.1 

4-6 

1875. 

25 

—  12.2 

—     1.0 

-13.8 

—     I  .2 

Jan.       I 

-    8.7        +     1.2 

-   7-9 

+   4-1 

26 

12.5 

0.6 

14.1 

4-    0.5 

2 

8.2                I.I 

7-9 

3-5 

27 

12.6 

-  0.3 

13.8 

2.0 

3 

7-8 

I  .0 

7.'> 

2.9 

28 

12.5 

0.0 

13.0 

3.3 

7-4 

0.9 

7-9 

2.1 

29 

12.1 

+  0.4 

11.9 

4-3 

5 

7-1 

0.6 

7-8 

1.2 

30 

-II.  7 

+  0.7 

—  10.9 

+    4-9 

6 

-  6.  Q 

+  0.4 

-  7-8 

4-   0.4 

Dec.      I 

II.  0 

0.9 

9.9 

5-1 

7 

6.8 

+    0.2 

7-7 

-  0.5 

2 

10.3 

I.I 

9.0 

5-i 

8 

6.9 

O.O 

7-5 

"•4 

3 

9-4 

1.2 

8.2 

4.7 

9 

7-2 

-  0.3 

7-4 

2.2 

4 

8.5 

I  .2 

7.5 

4.2 

10 

7-6 

0.5 

7-3 

3-0 

5 

-   7-7 

+     I.I 

-  7.0 

4-  3-6 

ii 

-  8.0 

-  0.7 

-  7-3 

-    3-7 

6 

7-0 

I.O 

6.8 

3.0 

12 

8.6 

I.O 

7-5 

4-3 

7 

6.4 

0.9 

6.6 

2.3 

13 

9.2 

1.2 

8.0 

4-7 

8 

6.1 

o.S 

6.5 

1.6 

14 

9-7 

1.3 

8.6 

4-9 

9 

6.0 

0.6 

6.7 

0.9 

15 

to.  3 

1.4 

9-5 

4.8 

10 

-  6.1 

+  0.4 

-  6.9 

+    O.I 

16 

—  10.8 

-   i.3 

—  10.6 

-  4-3 

ii 

6.4 

+    O.2 

7-' 

-  0.7 

17 

II    2 

1.2 

II.  7 

3-4 

12 

6.8 

0.0 

7-3 

1-5 

18 

ii.  5 

I  .0 

12.6 

2.2 

'3 

7-4 

-  o-3 

7-5 

2.4 

19 

ii.  6 

0.8 

'3-1 

-  0.8 

14 

8.0 

0.6 

7-6 

3-3 

20 

ir.fi 

0.5 

13.0 

4-  0.8 

15 

-  8.7 

-  0.8 

-   7-8 

-  4.0 

21 

-ii.  5 

—  •O.I 

-12.4 

4-    2.2 

16 

9-3 

i.i 

8.1 

4-7 

22 

ii.  I 

+    0.2 

ii.  4 

3.3 

17 

9.9 

I  .2 

8.6 

5.0 

23 

10.7 

0.5 

10.3 

4-1 

18 

10.3 

1.3 

9.2 

5-1 

24 

IO.2 

0.7 

9-3 

4-5 

19 

10.7 

1-4 

IO.O 

4.8 

25 

9.6 

I.O 

8-5 

4.6 

20 

—  II.  0 

-   1-3 

—  II.  0 

-  4-1 

26 

-  9.1 

+   i.i 

-  7-9 

4-  4-6 

2 

II.  2 

1.2 

12.  0 

3-1 

27 

8.5 

i.i 

7-5 

4-3 

2 

II.  4 

1.0 

12.7 

1.7 

28 

8.1 

1.  1 

7-2 

4-0 

2 

II.  4 

0.7 

12.9 

—    0.2 

29 

7-7 

i.i 

7-2 

3-5 

2 

II.  4 

0.4 

12.6 

+   1-3 

30 

7-5 

I.O 

7-4 

2.9 

2 

-11.3 

—    O.I 

—  12.0 

+    2.6 

3' 

-  7-4 

+  0.9 

-   7.7 

4-   2.3 

2 

II.  I 

+  0.3 

II.  I 

3-6 

TABLES. 


T.\  I;  I.  I.S. 


TAIH.I    1 

TM...K  II 

/  \ilutt  i/  tkt  Ar^umcnti  for  tkt  /•. 

KtifuftitiH  nf  Ikf  Arguments  la  tkf  tent- 

Hing  of  fafk  i 

itey  of  fiit  A  month. 

/> 

H 

M 

lib. 

t 

D 

A 

/» 

• 

iSso 

1.8 

16.7 

»-J       43 

«   5 

Jan.      «>• 

o.o 

o.o 

o.o 

o.o 

o.o 

1851 

8.6 

•7-3 

o.o     12.7 

9-S 

Feb.      o* 

3.4 

1-S 

14-9       3-6 

38 

iSjt  B 

16.4 

9-4 

10.9     aa.i 

ai.7 

Mar.     o 

3-9 

0.0 

10.6       4.1 

4-6 

1853 

«3.l 

20.0 

4-7 

3  « 

5-8 

April    o 

7-3 

1.4 

9-J 

7-7 

8.4 

•  8$4 

»-4 

l.'l 

U-6 

u  .5 

17-0 

May      o 

9.8 

1. 

7-0 

10.3 

11.2 

llM 

9m 

11.8 

• 

|  _o 

•  055 

1856  B 

•  • 

17.0 

t3-4 

•  * 
3-» 

"I-  V 
•  9 

"3- 

July      o 

•S-7 

3- 

,, 

"7-7 

•  857 

»3  8 

4-S 

13.1 

10.3 

<4- 

AUK.    o 

19.1 

r. 

i.a 

ao.o 

21.5 

1858           30     15.1 

6.9     18.7 

8. 

Sept.     o 

22.6 

6. 

0.9 

as.  6 

25.2 

1859         9.8    as-8 

0.7     «7-l 

19. 

Oct.      o 

f|.0 

7-* 

•4-8 

26.1 

0-9 

iMoB 

.7-6 

7-9 

11.6       9.1 

4- 

Nov.     o 

Dec        O 

0.9 

4      1 

•  •7 

9t 

13  5 

111 

»  3 

•  • 

47 

v  • 

186* 

3.6     2<>.2     15.  a     as.  9 

O.I 

LTV.V.             V               J'J 

*  ' 

•  •••          •»  •  -          t  •  y    i 

1863        10.4  |  10.  a 

9.0       6.9 

11.3 

*ln  January  and  February  of  Icap-yeara. 

1864  B    18.3     21.9 

37     16.3 

a3.6 

(he  number*  taken  from  Table  II  arc  lo  be 

1865 

s$.o 

3-0 

«3-7     14.7 

7-6 

diminished  by  a  unit. 

1866 

4-* 

13.6 

7-4       5-7 

18.8 

1867 

II.  0 

*4-3 

i.a     14.1 

a.  8 

TAHI.K 

III. 

1868  B 
1869 

18.8 
•$.6 

6-4 
•7-0 

12.1     »3.5 
5-9       4-4 

15.1 
a6.3 

Qerwdt  of  the  Arguments. 

1870 

4.8 

»7.6 

IS.  8      12.8 

10.4 

•  871 

11.6 

8.7 

9.6     si.  3 

21.6 

t 

D 

A 

B 

• 

187*  B 

•9-4 

ao.4 

4.4       3-* 

6.6 

•  873 

26.2 

i.S 

14-3      ««•* 

"7-9 

P  .    . 

a7-6 

»9-5 

16.1 

•7-4 

*7-» 

1874 

5-4 

ia.1 

8.0     20.0 

1.9 

2    /'    .      . 

55-1 

59.1 

3*  3 

$49 

54  4 

1875 

11.  » 

22.7 

1.8 

I.O 

13.1 

3  r  .   . 

8a.7 

88.6 

48.4 

82.3 

ll.6 

1876  B 

•0.0 

4-8 

ia.7 

10.4 

*5  4 

4  r  .   . 

no.  a 

118.1 

64.6 

109.7 

108.8 

1877 

26.8 

IS-  5 

6-5 

18.8 

9-4 

1878 

6.0 

26.1 

o-3 

27.  a 

•0.7 

. 

• 
1880  B 

ao.6 

18.8 

S-o 

17.6 

16.9 

1881 

«7-4 

»9-4 

14-9 

•6.0 

l.o 

1882 

6.6 

10.6 

8.6 

7-0 

12.  a 

'  1883 

U  4 

ai.a 

2.4 

«5-4 

»3  4 

1884  B 

tt.a 

3-3 

13  3 

a4-8 

«  5 

1685 

0.5 

13-9 

7-« 

5.« 

•9-7 

1886 

7-3 

84-6 

0.9 

14.2 

3-7 

1887 

M.O 

$-7 

10.8 

22.6 

15.0 

1888  B 

ai.8 

"7-3 

55 

4-6 

o.o 

1889 

28.6 

*7-9 

"5-4 

13.0 

u.  a 

46 


TABLE  IV. 

Secular  Terms. 

TABLE  V. 

Argument  A. 

TABLE  VI. 

Argument  B  (Empirical  Term}. 

Year.            n6z              Diff. 

A 

» 

B 

« 

B               fiv 

1848.0 

o.oo 

0 

0.00 

0 

o.oo 

40     .     4-0.39 

4-    0.20 

1849.0 

4-   O.2O 

i           —  0.23 

i 

+  0.34 

41           4-  0.05 

1850.0 

O.2O 
0.4O 

2 

-  0.44 

2 

0.66 

42           —   0.29 

1851.0 

0.6O 

0.20 

3 

-  0.57 

3 

0.95 

43           -  o  62 

O.2O 

4           —  0.62 

1852.0 

0.80 

4 

1.19 

4-1           -   0.91 

O.20 

5            —  0.57 

1853.0 

I.OO 

4-    O.2O 

6            -  0.44" 

5 

+   1-37 

45           -    1.16 

1854.0 

4-    1.  2O 
0.20 

7            -  0.25 

6 

'•47 

46           -    1.34 

1855.0 

I    4O 

8           —  0.02 

7 

1.50 

47           -    1.46 

4-  o.io 

1856.0 

1.50 

g            4-  0.22 

8 

1-45 

48           -    1.50 

o.oo 

1857.0 

1.50 

10                0.42 

9 

1.32 

49           -    1.46 

1858.0 

o.oo 

1.50 

II                0.56 

10 

4-    1.13 

50           -    1-34 

o.oo 

12                     O  62 

1859.0 

4-    1.5° 

ii 

0.88 

51           -    1.16 

O.OO 

13    !         0.58 

1860.0 

1.50 

O.OO 

14            4-  0.46 

12 

0.57 

52           -  0.91 

1  86I.O 

1.50 

O.OO 

15 

0.26 

13 

4-  0.25 

53     ,      -   0.62 

1862.0 

1.50 

—    O.O3 

16            4-  0.03 

14 

-    0.  10 

54     '      -  0.29 

1863.0 

1-47 

-    1.  12 

17               —    O.2O 

15 

-  o-44 

55     :     +  0.05 

1864.0 

4-  0.35 

18     i        -  0.42 

16 

-  0.75 

56              0.39 

1865.0 

-  0.73 

-    1.  08 

19           —  0.56 

1  "* 

-   1.03 

57                0.71 

1866.0 

-    1.77 

-    1.04 

20 

—  0.62 

11 

-    ".25 

58                0.99 

1867.0 

-   2.77 

-    I.OO 

21 

-  0.59 

"9 

-    1.40 

59                    1.22 

—  0.96 

22 

-  0.47 

1868.0 

-  3-73 

—   0.92 

23 

-  0.28 

20 

-    1.49 

60          4-    1.39 

1869.0 

-  4.65 

-  0.88 

24                -    O.O5 

21 

-    1-49 

61               1.48 

1870.0 

-  5-53 

-  0.84 

25               4-    O.lg 

22 

1.42 

62               i  .  50 

1871  .0 

-6.37 

—  0.80 

26                      0.40 

23 

-    '.27 

('3                1-44 

1872.0 

-   7-'7 

27 

0.55 

24 

-    1.  06 

64                l  .  30 

1873.0 

-   7-93 

—  o.  76 

2.8                      0.62 

25 

-   0.79 

65          4-    i.  08 

1874.0 

—  0.72 
-  8.65 

29                0.59 

26 

-  0.48 

66               0.83 

1875.0 

-  9-33 

—    0.08 

30            4-  0.49 

27 

-  0.15 

6?               0.53 

1876.0 

-  9-97 

-    0.64 
—    O.6o 

31                  0.30 
32            4-  0.07 

28 

4-   0.19 

68          4-  o.  19 

1877.0 

-  10.57 

n  56 

33            -  0.17 

29 

0.53 

69          —0.15 

1878.0 

—  11.13 

—  0.52 

34 

-  0.38 

30 

4-   0.83 

70          -  0.48 

1879.0 

—  11.65 

-   0.48 

35 

-  o.54 

3' 

1.  08 

71            -   0.79 

1880.0 

—  12.13 

36 

—  0.62 

32 

1.30 

72            -    1.  06 

37 

—  0.60 

33 

i  .44 

77            —    I    27 

38 

-  0.49 

34 

1.50 

/  J                      '  *  / 
74            -    1.42 

39 

-  0.31 

35 

4-    1.48 

75           -    '-49 

40 

—  0.09 

36 

'•39 

76           -    1.49 

41 

4-  0.15 

42 

0.38 

37 

1.22 

77           -    1.40 

43                 0.54 

38 

0.99 

78          -    1.25 

44                 0.61 

39 

0.71 

79          -   1.03 

45                 0.60 

40 

4-  0.39 

-  °-75 

46 

4-  0.51 

*tw 

47 

0.33 

48 

4-  o.io 

49 

-  0.14 

50 

—  0.36 

47 

TAIL.    \  II,  N**. 
Argu*ifHti}  1)  ./«./  Ihf  msnth. 


• 

Mai. 

April.        May. 

June* 

July.        Aug.        Sept 

-O.OI    > 

-O.OJ 

-0.04 

l  J  -0.03 

-O.OI 

+  o  01      +0.03 

+  0.04        +0.04 

+    O.OJ       + 

1 

+O.OI        +0.01 

+  0.03     +0.0$ 

+0.08      +0.10      fo.ii      +0.11      +0.10 

+  0.07      +  0.0$ 

J 

0.09        o.io 

O.  IJ         0.16 

O.I91        0.31            O.JO           O.I9 

0.16 

0.13          o.io 

3 

0.19        o.n 

0.16 

0.39 

11.33          0.34          0.31          O.JS 

o.ll  '       0.18 

i          0.19 

0.33         o  38 

0.41 

0.49  '     0.49       0.46       0.41 

0.39 

<         *o.4$ 

+o.$o      +o.$6 

+  0.61      +0.66 

+0.67  1    +o  67      +0.61      +o.$6 

0.4$ 

6           0.63 

0.68         0.77 

0.83         0.8$ 

0.87 

0.8$ 

0.79 

0.71 

o.6j           0.60 

7          0.80 

0.87  I      0.96 

1.01            1.0$ 

1.01 

1.  01 

0.86 

0.80 

0-77 

8           0.93 

1.09             1.13 

1.18  '       i.jo 

II,        II           1 

0.90 

0.88            0.89 

9           1.01 

1.12              I.JI 

1.37  !       1.39 

1.37             1.31 

l.ll             I.OI 

0.9$ 

o.ob 

10        +I.«>4 

+  1.1$        +1.1$ 

+  1.30  j   +1.30 

+  1.36      -t-l.lS 

+  1.08     +0.98 

il       4    0.97 

II             0.97 

1.07              I.  It 

1.13            1.10 

1.1$          1.07  '      0.96         0.86 

0.81 

0.83 

13             0.81 

0.9!              1.03 

1.06 

I.OJ 

0.96         0.86 

0.76 

0.63 

o.frfi            0.73 

13          o.$8 

o.6(         0.77 

"  ••) 

0.66         o.$6         0.46 

0.37 

0.35 

0.48 

14        +0.38 

0.39         0.47 

0.48  I     0.41 

+0.33 

+  0.11 

+0   11 

+0.04 

+0,03 

+  O.OQ 

+  0.18 

i$ 

+0.08      +0.14 

+0.14 

—0.03 

-O.IJ 

—0.13 

—O.JO 

-O.JO 

-   0.33 

16 

-0.34 

-0.3$       —0.30 

-  0.11 

-0.19 

-0.40 

—  o.$o 

-o.$9 

—0.64 

—0.61 

-  o.$$ 

-  0.44 

17 

-0.60 

-o.$J     -0.49 

-0.$J 

-0.61 

—0.71 

—0.81 

-0.89 

-0.01 

-0.89 

-  0.81 

-  0.70 

18 

—o.8l 

-0.74     -0.71 

—0.76 

-0.66 

-0.97 

-1.0$        -I.IJ 

-1.14 

—  1.00 

—    I.OO 

—  0.89 

19 

-0.93 

-0.88     -0.88 

—0.89     —I.OJ 

—  1.13       -1.31 

-1.16 

-1.13 

-1.11 

—    1.13        —    I.OI 

30 

-0.97 

—0.9* 

-0.04 

—  i.oo  1  —  l.io 

—  1.31       —1.17 

-I.JO 

-I.JO 

-1.14 

-    1.14    '    -    I.OJ 

31 

-0.93 

-0.91 

-0.93, 

-0.09 

-1.0, 

-1.18      -1.14 

-1.1$ 

-I.IJ 

-1.17 

-     1.0* 

-  0.99 

33 

-0.83 

-0.83 

-0,86 

-0.91 

—  I.OI 

-1.09      -1.13 

-1.13 

—  I.  10        - 

-  0.9$ 

-  0.87 

33          -0.69 

—0.70 

-0.71 

-0.78 

-0.87 

-0.94 

-0.96 

-0.0$ 

—0.91      —0.86 

-0.78 

—  0.67 

24           ~0.$l 

-0.$4 

-o.$9 

—0.64 

-0.71 

-0.77 

—0.79     —0.77 

-o.7J 

-0.67 

—  0.60 

-  0.5$ 

35           -0.37 

-0.39 

-0.43 

—0.48 

-0.$4 

-0.59 

—o.  $9      —o.  $6 

-0.$J 

-0.47 

-  04' 

-  0.37 

16    ]  -0.13 

-0.17 

-0.31 

-0.36 

-0.40 

-0.41 

—0.41      —  O.j8      —  0.34 

—0.19 

—    0.1$ 

17        -0.14 

—0.17 

—o.jo 

—0.34 

-0.17 

—0.18     -0.16     -0.33     -o.io 

—  o.  16 

-O.IJ 

—    O.ll 

•8    '  -0.07 

—  0.  II 

—o.ll 

1    -0.1$ 

—  o.  16 

—  o.  16     -0.14 

—o.io     —0.09 

—0.06 

-    0.0$ 

-    0.0$ 

•9        -0.03 

—0.06 

—0.07 

-0.08 

-0.08 

—0.06 

-0.04 

—O.OJ 

0.00 

+0.01 

o.oo 

jO 

o.oo 

—O.OI 

—O.OI 

—O.OI 

o.oo 

+0.01     +0.04 

+o  06     +0.07 

+   0.0$    ;    +    0.01 

3« 

+0.0$ 

+0.0$ 

+0.07 

+0.08 

+o.  10 

0.13        0.1$ 

0.16           0.14 

0.13 

O.ll 

0.07 

3» 

+0.13 

+0.14 

+0.16 

+0.19 

+  0.33 

+0.16 

+0.17 

+0.16 

+  0.1J 

+  0.30 

+    0.17 

+  0.14 

Nont.— Each  column  is  computed  for  Ihc  middle  of  ihc  month,  but  may  be  used  for  the  cnlifc  month  without  an 
error  crer  exceeding  o."o«.     If  much  greater  accuracy  than  this  is  required,  a  boritonial  interpolation  mu»t  be  used. 


48 
TABLE  VIII,  Sv. 

Horizontal  Argument,  or  Argument  at  top,  -D—g,  or  D— §"+30.      Vertical  Argument,  g. 


g 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

IO 

II 

12 

'3 

14 

0 

—0.23 

—0.30 

-0.36 

-0-39 

-0-39 

-0.35 

—  0.2S 

—  o.  19 

—  0.  11 

—0.03 

+O.O2 

+0.03 

O.OO 

—  O.O6 

-0.15 

I 

-o.39 

—  0.46 

—  0.50 

-0.53 

—0.50 

-0.44 

—0.36 

—0.28 

—  o.  19 

—  O.I2 

-O.Og 

-O.Og 

-0.14 

—  O.22 

-0.31 

2 

-0.54 

—O.6o 

—  0.64 

—0.62 

-0.58 

-0.52 

-0.43 

-0-34 

—0.25 

—  O.2O 

—  o.  19 

—  0.21 

-0.27 

-0.35 

-o  45 

3 

-0.66 

-0.71 

—0.72 

—0.69 

—0.64 

-0.56 

-0.47 

-0.37 

-0.31 

—0.27 

—  0.26 

-0.31 

-O.J8 

-0.47 

-0.58 

4 

-0.75 

-0.77 

-0.77 

-0-74 

—0.67 

-0.58 

—0.48 

-0.39 

-0.34 

—  0.31 

-o.33 

—0.38 

—  0.46 

-0.57 

—0.67 

5 

-0.79 

—  0.81 

—0.80 

-0.74 

-0.66 

-0.55 

—0.46 

-0-39 

-0.34 

-0.34 

—0.36 

-0.43 

-0.53 

—  0.64 

-o.74 

6 

—0.80 

—  0.81 

—  0.76 

—0.70 

—0.60 

—0.50 

—0.42 

-0.35 

-0-33 

-0.33 

-0.38 

-0.47 

-0.57 

—0.67 

—0.76 

7 

-0.78 

—0.76 

-0.71 

—  0.62 

—0.52 

-0.43 

-0.34 

—0.30 

—0.28 

—O.3I 

-0.38 

-0.47 

-0.57 

—0.67 

-o.73 

8 

-0.71 

-0.6S 

—  0.61 

-0.51 

—  0.    2 

—0.32 

-0.26 

—  O.22 

—  O.22 

—O.27 

-0.35 

-0.45 

-0.54 

—  0.61 

-0.63 

9 

—0.62 

—0.56 

-0.47 

-0.38 

—0.28 

—  O.2O 

-0.14 

—  O.I2 

-0.15 

—  O.2I 

—0.30 

—0.38 

-0.47 

-0.53 

-0.58 

10 

-0.49 

-0.41 

-0-33 

—  O.22 

—0.13 

—O.O6 

—  O.O2 

—  O.O2 

—  O.o6 

-0.13 

—  0.22 

—0.30 

-0.37 

-0.44 

—0.48 

II 

-0.33 

—0.26 

—  o.  16 

—O.O6 

+0.03 

+0.09 

4-O.II 

+O.  IO 

+0.04 

—0.03 

—  O.I2 

—0.19 

—0.28 

-0.33 

—0.33 

12 

—  0.18 

—0.08 

+O.O2 

+0.  II 

0.19 

O.24 

0.25 

0.21 

0.15 

H-o.07 

—0.01 

—  O.IO 

-0.17 

—  0.18 

—  0.18 

'3 

o.oo 

+O.IO 

0.19 

0.28 

o.35 

0.38 

o.37 

o.33 

0.26 

0.19 

+  0.09 

o.oo 

-0.03 

—0.04 

—  O.OI 

M 

-1-0.17 

0.26 

0.36 

0.45 

0.50 

0.51 

0.49 

0.41 

0.37 

0.27 

0.17 

+0.13 

+0.  IO 

+O.II 

+0.14 

'5 

+0.32 

+0.42 

+0.52 

+0.59 

+0.63 

+  0.62 

+0.59 

-1-0.53 

+  0.43 

+0.34 

+  0.28 

4-0.23 

+0.23 

+0.24 

4-0.29 

16 

0.46 

0.56 

0.65 

0.71 

0.73 

0.71 

0.66 

0.58 

0.48 

0.42 

0.35 

0.33 

0.33 

0.37 

0.43 

17 

'  0.58 

0.68 

0.75 

0.80 

0-79 

0.76 

0.68 

0.60 

0-53 

0.46 

0.42 

0.41 

0.43 

0.48 

0.56 

18 

0.67 

0.76 

0.82 

0.84 

0.82 

o.77 

0.69 

0.62 

0.54 

0.49 

0.46 

0.47 

0.50 

0-57 

0.65 

'9 

0-73 

0.80 

0.83 

0.84 

0.81 

0-75 

0.67 

0.59 

0-53 

0.48 

0.48 

0.49 

0.56 

0.63 

0.72 

20 

+0.74 

+0.79 

+O.SI 

+0.80 

+0.76 

+0.69 

+0.61 

+  0.54 

4-0.49 

+  0.47 

4-0.46 

+0.51 

+0.58 

+0.67 

+0.73 

21 

0.70 

0.75 

0.76 

+  0.73 

0.67 

0.60 

0.53 

0.47 

0.43 

0.41 

0.44 

0.51 

0.58 

0.65 

0.71 

22 

0.64 

0.67 

0.67 

•HO.62 

0.56 

0.48 

0.42 

0.37 

o.33 

0-35 

0.40 

o.47 

0.53 

0.60 

0.67 

23 

0.55 

0-57 

0.54 

+  0.49 

0.42 

0.35 

0.29 

0.24 

0.24 

0.27 

0-33 

0.38 

0.48 

0-55 

0.60 

24 

0.45 

o.43 

0.39 

+  0.33 

0.26 

0.20 

-1-0.13 

4-O.  !2 

0.13 

0.17 

O.22 

.0.31 

0.40 

0.46 

0.50 

25 

+0.29 

+0.28 

+0.22 

+0.16 

+O.IO 

+  O.02 

—O.OI 

—  O.O2 

+O.OI 

4-0.05 

+0.14 

4-0.24 

+0.31 

+0.36 

+  0.37 

26 

+1.15 

4-O.II 

+0.06 

—  O.OI 

—O.og 

—  O.I4 

—o.  16 

—  o.  16 

-0.13 

—0.04 

4-O.06 

0.13 

O.20 

0.23 

0.23 

27 

O.O2 

—0.06 

—  O.I2 

—  O.20 

—O.26 

—  O.30 

-0.31 

—  0.30 

—0.22 

-0.13 

—O.O4 

+0.04 

+0.08 

+  0.  10 

4-0.  08 

28 

-0.17 

—  0.22 

—  O.3O 

-0.37 

—0.42 

—  0.46 

—0.46 

-o.39 

-0.31 

—  0.22 

-0.13 

—0.07 

—  O.O2 

—0.03 

—  0.07 

2J 

-0.31 

-0.38 

—  0.46 

-0.52 

-0.57 

-0.59 

-0.54 

-0.47 

-0.39 

—  0.29 

-0.22 

-0.15 

-0.13 

-0.15 

—  O.2O 

30 

-0.45 

—O.52 

-0.60 

-0.66 

—  O.6g 

-0.66 

—0.60 

-0.54 

-0.44 

-0.36 

—O.27 

—0.23 

—  0.23 

—0.26 

—0.32 

I'.t 

1  \    i      \  I  II      ---  Ciiiiliiin.-.l 

+  JO.         / 


o 
I 

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• 

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3 

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-o.6l 
-0.7*. 

-0.7* 

-0.78 
.  -0.81 

-0.67 

—  o.ta 

S 

—  0.81 

o.$» 

6 

-0.80 

• 

-o  47 

-0.6o 

7 

-0.77 

-0.42 

8 

-0.6, 

—0.63 

9 

ii 

—  o  60 
-0.46 

-O.JO 

-0.41 
—0.35 

-0.38 

• 

-O.IJ 

t  0.  12 

o.oo—  o.oq 

13 

+0.03 

—  0  Ofl 

'i 

0.30 

o.s« 

+0.37 

*-o.si 

4-O.5J 

'7 

0.51 

0.63 

0.65 
0.7-1 

0.6, 

0.31 

o.3i 

0.78 

o.Tt 

0.77 

0.80 

0.81 

•     o.  jt 

• 

+0.77 

+0.80 

+  o  6ft 

1 

0.76 

0.76 

o.t» 

0.70 

0.68 

0.55 

•  •  t^ 

0.30 

o.6» 

0.60 

o.S» 

0.51 

+0.  It 

*4 

• 

o.tb 

*S 

fO.JJ 

+o.»7 

-t-o.lq 

+0  O-i 

O.OU 

-0.11 

36 
*7 

o.lS 
4-0.01 

—  0.061 

+0.01 
-0.lt 

-0.08 

1 

-o.M 

-0.14 

-0.3* 

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3» 

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. 

7  M 


50 


TABLE  IX. 

Argument,  g.  factor  to  be 
multiplied  by  n  Sz. 

TABLE  X. 

Argument,  u.  Factors  for  correction  of  latitude  and  reduc- 
tion to  ecliptic  longitude. 

'_ 

F 

u                  (F-l) 

WJ 

« 

o 

+  0.118 

o 

-    O.OO4 

+  o.ogo 

o.oo 

i 

0.114 

i 

-    O.OO4 

0.088 

—  0.03 

2 

0.103 

2 

-    O.OO3 

0.081 

—  0.07 

3 

0.086 

3 

—    O.OOI 

0.069 

—    O.  IO 

4 

0.065 

4 

+    O.OOI 

0.054 

—    0.12 

5 

+  0.040 

5 

+  0.003 

+  0.036 

—  0.14 

6 

•+  0.015 

6 

0.004 

+  0.017 

-  0.15 

7 

—  0.009 

7 

0.004 

—  0.004 

-  0.15 

8 

-  0.034 

8 

0.004 

-  0.024 

-  0.14 

9 

-  0.054 

9 

+    O.OO2 

-  0.044 

-  0.13 

10 

—  0.072 

10 

O.OOO 

—  0.060 

—   0.11 

1  1 

—  0.086 

ii 

—    O.OOI 

-  0.074 

—  0.08 

12 

—  0.096 

12 

—  0.003 

-  0.084 

—  0.05 

13 

—    O.IOI 

13 

-  0.004 

—  0.089 

—   O.O2 

M 

-  0.103 

'4 

—  0.004 

—  0.089 

+    0.01 

15 

-  0.099 

15 

—  0.003 

—  0.085 

+  0.05 

16 

—  0.092 

16 

—    O.OO2 

—  0.076 

0.08 

17 

—  0.080 

17 

O.OOO 

—  0.064 

0.11 

18 

—  0.065 

18 

+    O.OO2 

-  0.047 

0.13 

19 

—  0.046 

'9 

O.OO3 

—  0.028 

0.14 

20                       -    0.024 

20 

+    O.OO4 

—  0.008 

+  0.15 

21 

+    O.OOI 

21 

O.O04 

+   O.OI2 

0.15 

22 

0.026 

22 

0.003 

0.032 

0.14 

23 

0.051 

23 

+    O.OOI 

0.050 

0.12 

24 

0.075 

24 

O.OOO 

0.066 

O.IO 

25 

4-  0.094 

25 

—   O.OO2 

+  0.078 

+  0.07 

26 

o.  109 

26 

-    O.OO4 

0.086 

0.04 

27 

0.116 

27 

-    O.OO4 

o.ogo 

+    O.OI 

28 

0.117 

28 

—   O.OO4 

0.089 

—  0.03 

29 

O.IIO 

29 

—    0.003 

0.082 

—  0.06 

30 

+  0.096 

30 

—    O.OOI 

+  0.072 

-  0.09 

TABU  \l 


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